# Encyclopedia

A definite integral of a function can be represented as the signed area of the region bounded by its graph.
.Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus.^ Differentiability and the main theorem of the differential calculus.

^ This book presents 35 activities for middle school students that integrate the teaching of mathematical concepts with environmental concepts.

^ Integration of rational functions by partial fractions is a fairly simple integrating technique used to simplify one rational function into two or more rational functions which are more easily integrated.
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

Given a function ƒ of a real variable x and an interval [ab] of the real line, the definite integral
$\int_a^b \! f(x)\,dx \,$
is defined informally to be the net signed .area of the region in the xy-plane bounded by the graph of ƒ, the x-axis, and the vertical lines x = a and x = b.^ A second students´ profile is associated with those who apply successfully the idea of approximation to determine areas of bounded regions.
• Promoting Students´ Comprehension of Definite Integral and Area Concepts Through the Use of Derive Software 12 September 2009 12:49 UTC www.allacademic.com [Source type: Reference]

.The term integral may also refer to the notion of antiderivative, a function F whose derivative is the given function ƒ.^ An indefinite integral of a function f(x) is also known as the antiderivative of f.
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

^ Slightly more satisfactory answers -- i.e., more general notions of integral -- were given by Arnaud Denjoy (1912) and Oskar Perron (1914).
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ This property allows us to easily solve definite integrals, if we can find the antiderivative function of the integrand.
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

.In this case it is called an indefinite integral, while the integrals discussed in this article are termed definite integrals.^ This type of integral is called a definite integral .
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

^ This expression is called a definite integral.
• 4. The Definite Integral 12 September 2009 12:49 UTC www.intmath.com [Source type: FILTERED WITH BAYES]

^ We will introduce the definite integral defined in terms of area.
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

.Some authors maintain a distinction between antiderivatives and indefinite integrals.^ An indefinite integral of a function f(x) is also known as the antiderivative of f.
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

^ Because the integral is indefinite, we are left with some arbitrary C (a constant of integration,) because when the derivative of sin(x) + C is taken, C is insignificant to the result.
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^ Saks mentioned in his book that you can approach an integral descriptively (like Newton -- e.g., the integral is some sort of antiderivative) or constructively (like Riemann -- e.g., the integral is some sort of limit of approximating sums).
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century. Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if ƒ is a continuous real-valued function defined on a closed interval [ab], then, once an antiderivative F of ƒ is known, the definite integral of ƒ over that interval is given by
$\int_a^b \! f(x)\,dx = F(b) - F(a)\,$
.Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering.^ MATH 145 Calculus for Life Sciences (5) NW, QSR Curtis, Smith, Tuncel Differential and integral calculus, with examples from the life sciences.

^ Emphasis on these tools to formulate models and solve problems arising in variety of applications, such as computer science, biology, and management science.

^ MATH 146 Calculus for Life Sciences (5) NW Curtis, Smith, Tuncel Further applications of integration; elementary differential equations, with examples from the life sciences.

.A rigorous mathematical definition of the integral was given by Bernhard Riemann.^ In fact, the Henstock-Kurzweil formulation -- the gauge integral -- is considerably simpler than the Lebesgue idea, and its definition is only slightly different from the definition of the Riemann integral.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ To put the gauge integral into proper perspective, we should first review the definition of the Riemann integral.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ Bartle's article, "Return to the Riemann integral," in the October 1996 American Mathematical Monthly , is brief and easy to read; probably it was read widely.
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.It is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs.^ By inserting new vertices into the edges and faces of a hypercube, it may be "inflated'' to give an approximation to the hypersphere.
• Dr. Alice Christie's Technology Integration: Math 15 September 2009 16:20 UTC www.west.asu.edu [Source type: FILTERED WITH BAYES]

^ In the first group, students, in general, recognize the importance of finding areas of limited curves through the idea of approximation.
• Promoting Students´ Comprehension of Definite Integral and Area Concepts Through the Use of Derive Software 12 September 2009 12:49 UTC www.allacademic.com [Source type: Reference]

^ Well, recall that we can always estimate the area by breaking up the interval into segments and then sketching in rectangles and using the sum of the area all of the rectangles as an estimate of the actual area.
• http://tutorial.math.lamar.edu/Classes/CalcII/IntegralTest.aspx 14 January 2010 19:22 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

.Beginning in the nineteenth century, more sophisticated notions of integrals began to appear, where the type of the function as well as the domain over which the integration is performed has been generalised.^ Collectively, we have kissed a lot of frogs, and are now ready to design and construct transformational systems that more predictably awaken and stabilize integral functioning.

^ Slightly more satisfactory answers -- i.e., more general notions of integral -- were given by Arnaud Denjoy (1912) and Oskar Perron (1914).
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ (Perhaps we should deemphasize the other measures in the first graduate course on integration theory, in order to make the gauge integral fit into the course more readily.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

.A line integral is defined for functions of two or three variables, and the interval of integration [ab] is replaced by a certain curve connecting two points on the plane or in the space.^ A curve is made by connecting the points on the graph.
• The ABCs of Pharmacokinetics - The Body 20 September 2009 1:01 UTC www.thebody.com [Source type: Academic]

^ The real numbers as a complete ordered field, infinite sequences of real numbers, real valued functions of a single real variable: limits and continuity, continuity on a closed interval, monotonic functions, inverse functions, differentiability and the fundamental theorem of differential calculus, Taylor's theorem, L'Hopital's rule, curve tracing, elementary, functions, methods of integration, definite integrals, integrable functions, fundamental theorems of integral calculus, improper integrals.

^ It can be generalized to functions defined on the whole real line, or to functions defined on finite-dimensional spaces, but its definition and theory then become slightly more complicated.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

.In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space.^ Explore four-dimensional structures, projected into three-dimensional space.
• Dr. Alice Christie's Technology Integration: Math 15 September 2009 16:20 UTC www.west.asu.edu [Source type: FILTERED WITH BAYES]

.Integrals of differential forms play a fundamental role in modern differential geometry.^ Offered: A. MATH 442 Differential Geometry (3) NW Curves in 3-space, continuity and differentiability in 3-space, surfaces, tangent planes, first fundamental form, area, orientation, the Gauss Map.

^ The ideas embedded in this framework played a fundamental role not only in analysing students´ work but also influenced the design and structure of the study.
• Promoting Students´ Comprehension of Definite Integral and Area Concepts Through the Use of Derive Software 12 September 2009 12:49 UTC www.allacademic.com [Source type: Reference]

^ The Integral represents a fundamentally new stage in evolution, and a fundamentally new form of humanity .
• Integral Paradigm 101 14 January 2010 19:22 UTC www.audettesophia.com [Source type: FILTERED WITH BAYES]

.These generalizations of integral first arose from the needs of physics, and they play an important role in the formulation of many physical laws, notably those of electrodynamics.^ These do not represent all possible Integral/GCN transmissions only those we choose to receive .

^ These groups have different visions about what integral is all about and those visions should be spelled out; but ARINA and Polysemy need not agree with each other.
• What does the Integral Movement represent? « Open Integral 14 January 2010 19:22 UTC www.openintegral.net [Source type: Original source]

^ Many important functions do not have a Riemann integral -- even after we extend the class of integrable functions slightly by allowing "improper" Riemann integrals.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

.There are many modern concepts of integration, among these, the most common is based on the abstract mathematical theory known as Lebesgue integration, developed by Henri Lebesgue.^ Conceptual Framework Heid (2002) recognized that the use of CAS has influenced the development and refinement of theories that explain the students´ learning of mathematics.
• Promoting Students´ Comprehension of Definite Integral and Area Concepts Through the Use of Derive Software 12 September 2009 12:49 UTC www.allacademic.com [Source type: Reference]

^ There are many ways to categorize the various types of systems theory, from historical to methodological to theoretical.
• Ken Wilber Online: Excerpt C - The Ways We Are in This Together 29 September 2009 17:46 UTC wilber.shambhala.com [Source type: FILTERED WITH BAYES]

^ This book presents 35 activities for middle school students that integrate the teaching of mathematical concepts with environmental concepts.

## History

### Pre-calculus integration

Integration can be traced as far back as ancient Egypt ca. 1800 BC, with the Moscow Mathematical Papyrus demonstrating knowledge of a formula for the volume of a pyramidal frustum. .The first documented systematic technique capable of determining integrals is the method of exhaustion of Eudoxus (ca. 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of shapes for which the area or volume was known.^ The definite integral is defined to be exactly the limit and summation that we looked at in the last section to find the net area between a function and the x -axis.
• http://tutorial.math.lamar.edu/Classes/CalcI/DefnofDefiniteIntegral.aspx 14 January 2010 19:22 UTC tutorial.math.lamar.edu [Source type: Reference]

^ We need to figure out how to correctly break up the integral using property 5 to allow us to use the given pieces of information.
• http://tutorial.math.lamar.edu/Classes/CalcI/DefnofDefiniteIntegral.aspx 14 January 2010 19:22 UTC tutorial.math.lamar.edu [Source type: Reference]

^ In the first group, students, in general, recognize the importance of finding areas of limited curves through the idea of approximation.
• Promoting Students´ Comprehension of Definite Integral and Area Concepts Through the Use of Derive Software 12 September 2009 12:49 UTC www.allacademic.com [Source type: Reference]

.This method was further developed and employed by Archimedes and used to calculate areas for parabolas and an approximation to the area of a circle.^ Thus, the use of the Utility File could help students develop an image of the integration processes and its relationship with the area concept.
• Promoting Students´ Comprehension of Definite Integral and Area Concepts Through the Use of Derive Software 12 September 2009 12:49 UTC www.allacademic.com [Source type: Reference]

^ They were fluently not only in deciding what type of partition to take on the interval but also in using algebraic tools to carry out the operations involved in calculation the corresponding areas.
• Promoting Students´ Comprehension of Definite Integral and Area Concepts Through the Use of Derive Software 12 September 2009 12:49 UTC www.allacademic.com [Source type: Reference]

^ In particular, a special Utility File was designed to help students calculate approximations of areas.
• Promoting Students´ Comprehension of Definite Integral and Area Concepts Through the Use of Derive Software 12 September 2009 12:49 UTC www.allacademic.com [Source type: Reference]

.Similar methods were independently developed in China around the 3rd century AD by Liu Hui, who used it to find the area of the circle.^ Provides methods for cooperative, student investigation of weather data similar to methods currently used by atmospheric scientists.

^ For example, when finding the area of a circle or an ellipse you may have to find an integral of the form where a> 0.
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

^ Thus, the use of the Utility File could help students develop an image of the integration processes and its relationship with the area concept.
• Promoting Students´ Comprehension of Definite Integral and Area Concepts Through the Use of Derive Software 12 September 2009 12:49 UTC www.allacademic.com [Source type: Reference]

This method was later used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere.[1] .That same century, the Indian mathematician Aryabhata used a similar method in order to find the volume of a cube.^ Provides methods for cooperative, student investigation of weather data similar to methods currently used by atmospheric scientists.

^ Find locations on a map or grid using ordered pairs.

^ Similar to integrals solved using the substitution method, there are no general equations for this indefinite integral.
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

[2]
.The next major step in integral calculus came in Iraq when the 11th century mathematician Ibn al-Haytham (known as Alhazen in Europe) devised what is now known as "Alhazen's problem", which leads to an equation of the fourth degree, in his Book of Optics.^ Mostly integral calculus problems and problems with solutions .
• Comprehensive Ratings and Review of Calculus web sites 29 September 2009 17:46 UTC www.csam.montclair.edu [Source type: General]

^ Have you done those integral problems from the calculus book yet?
• Grace's work on the area under curves and the integral 20 September 2009 1:01 UTC www.mathman.biz [Source type: General]

^ Initial value problem for hyperbolic equations and methods of geometrical optics.

.While solving this problem, he performed an integration in order to find the volume of a paraboloid.^ Problem-solving should be a daily experience and should be integrated throughout the content areas.
• Benchmark Education Company - Math Content Integration 15 September 2009 16:20 UTC www.benchmarkeducation.com [Source type: FILTERED WITH BAYES]

^ An example to illustrate the type of problem in this group is, (ii) Problems in which students received an algebraic expression to find the integral and it was important to graph it in order to solve.
• Promoting Students´ Comprehension of Definite Integral and Area Concepts Through the Use of Derive Software 12 September 2009 12:49 UTC www.allacademic.com [Source type: Reference]

^ These properties of integrals of symmetric functions are very helpful when solving integration problems.
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

.Using mathematical induction, he was able to generalize his result for the integrals of polynomials up to the fourth degree.^ There are also some nice properties that we can use in comparing the general size of definite integrals.
• http://tutorial.math.lamar.edu/Classes/CalcI/DefnofDefiniteIntegral.aspx 14 January 2010 19:22 UTC tutorial.math.lamar.edu [Source type: Reference]

^ We will first need to use the fourth property to break up the integral and the third property to factor out the constants.
• http://tutorial.math.lamar.edu/Classes/CalcI/DefnofDefiniteIntegral.aspx 14 January 2010 19:22 UTC tutorial.math.lamar.edu [Source type: Reference]

^ In this case we’ll need to use Property 5 above to break up the integral as follows, .
• http://tutorial.math.lamar.edu/Classes/CalcI/DefnofDefiniteIntegral.aspx 14 January 2010 19:22 UTC tutorial.math.lamar.edu [Source type: Reference]

.He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.^ Find the general indefinite integrals 3 .
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

^ Find the general indefinite integrals using the substitution rule 7 .
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

^ However, the fact that we're using properties of intervals means that the gauge integral does not generalize readily to settings other than intervals.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

[3] .Some ideas of integral calculus are also found in the Siddhanta Shiromani, a 12th century astronomy text by Indian mathematician Bhāskara II.^ All of Differential Calculus, some Integral Calculus .
• Comprehensive Ratings and Review of Calculus web sites 29 September 2009 17:46 UTC www.csam.montclair.edu [Source type: General]

^ Comments: Site contains a list of quizzes for integral calculus, some quizzes have solutions.
• Comprehensive Ratings and Review of Calculus web sites 29 September 2009 17:46 UTC www.csam.montclair.edu [Source type: General]

^ Integral Calculus, Calculus-II .
• Comprehensive Ratings and Review of Calculus web sites 29 September 2009 17:46 UTC www.csam.montclair.edu [Source type: General]

.The next significant advances in integral calculus did not begin to appear until the 16th century.^ MATH 324 Advanced Multivariable Calculus I (3) NW Topics include double and triple integrals, the chain rule, vector fields, line and surface integrals.

At this time the work of Cavalieri with his method of indivisibles, and work by Fermat, began to lay the foundations of modern calculus. .Further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation.^ MATH 146 Calculus for Life Sciences (5) NW Curtis, Smith, Tuncel Further applications of integration; elementary differential equations, with examples from the life sciences.

^ Teachers who take advantage of opportunities to integrate vocabulary, strategy, and comprehension instruction into their mathematics lessons provide their students with a strong academic base.
• Benchmark Education Company - Math Content Integration 15 September 2009 16:20 UTC www.benchmarkeducation.com [Source type: FILTERED WITH BAYES]

^ The Fundamental Theorem of Calculus defines the relationship between the processes of differentiation and integration.
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

At around the same time, there was also a great deal of work being done by Japanese mathematicians, particularly by Seki Kōwa.[4] .He made a number of contributions, namely in methods of determining areas of figures using integrals, extending the method of exhaustion.^ This theme is a unique contribution of the integral yoga of Sri Aurobindo and The Mother and is consciously used by those practising integral yoga.
• Integral Paradigm 101 14 January 2010 19:22 UTC www.audettesophia.com [Source type: FILTERED WITH BAYES]

^ Similar to integrals solved using the substitution method, there are no general equations for this indefinite integral.
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

^ Lee Peng-Yee has given a Riemann-type definition of the Ito integral using the Henstock method.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

### Newton and Leibniz

.The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Newton and Leibniz.^ Antiderivatives and the fundamental theorem of calculus with applications.

^ With these formulas and the Fundamental Theorem of Calculus, we can evaluate simple definite integrals.
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

^ The Fundamental Theorem of Calculus defines the relationship between the processes of differentiation and integration.
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

.The theorem demonstrates a connection between integration and differentiation.^ The Fundamental Theorem of Calculus defines the relationship between the processes of differentiation and integration.
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. .In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems.^ Currently, WebMath can solve most problems from an Algebra I class.
• Dr. Alice Christie's Technology Integration: Math 15 September 2009 16:20 UTC www.west.asu.edu [Source type: FILTERED WITH BAYES]

^ With these formulas and the Fundamental Theorem of Calculus, we can evaluate simple definite integrals.
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

^ Students apply basic operations to problem-solving situations with a greater understanding of the meanings of operations and how they relate to one another.

.Equal in importance is the comprehensive mathematical framework that both Newton and Leibniz developed.^ Conceptual Framework Heid (2002) recognized that the use of CAS has influenced the development and refinement of theories that explain the students´ learning of mathematics.
• Promoting Students´ Comprehension of Definite Integral and Area Concepts Through the Use of Derive Software 12 September 2009 12:49 UTC www.allacademic.com [Source type: Reference]

^ But if properly understood and appropriately extended, they may be embraced within a framework that is at once true to their differences and yet comprehensive of both.

.Given the name infinitesimal calculus, it allowed for precise analysis of functions within continuous domains.^ Driven by Mathematica, it allows students to put in their own functions for analysis.
• Comprehensive Ratings and Review of Calculus web sites 29 September 2009 17:46 UTC www.csam.montclair.edu [Source type: General]

^ Even though physical science may continue indefinitely to expand the range of outer experience it comprehends, this expansion still takes place only within a limited domain of experience, namely, the outer sensory domain.

^ Offered: AWSpS. MATH 328 Introductory Real Analysis II (3) NW Limits and continuity of functions, sequences, series tests, absolute convergence, uniform convergence.

.This framework eventually became modern calculus, whose notation for integrals is drawn directly from the work of Leibniz.^ Categories: Inventions , Mathematics , People 1675: Gottfried Leibniz writes the integral sign ∫ in an unpublished manuscript, introducing the calculus notation that’s still in use today.
• Oct. 29, 1675: Leibniz ∫ums It All Up, Seriesly | This Day In Tech | Wired.com 14 January 2010 19:22 UTC www.wired.com [Source type: General]

^ Tags: Britain , calculus , France , Germany , Gottfried Leibniz , integral , Isaac Newton , simultaneous discovery Post Comment .
• Oct. 29, 1675: Leibniz ∫ums It All Up, Seriesly | This Day In Tech | Wired.com 14 January 2010 19:22 UTC www.wired.com [Source type: General]

^ The muddled back-and-forth eventually led to bad blood, with Newton claiming that Leibniz had stolen his work in founding the science of calculus.
• Oct. 29, 1675: Leibniz ∫ums It All Up, Seriesly | This Day In Tech | Wired.com 14 January 2010 19:22 UTC www.wired.com [Source type: General]

### Formalizing integrals

While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour. Bishop Berkeley memorably attacked infinitesimals as "the ghosts of departed quantities". Calculus acquired a firmer footing with the development of limits and was given a suitable foundation by Cauchy in the first half of the 19th century. .Integration was first rigorously formalized, using limits, by Riemann.^ Many use Jungian terminology to describe their growth process of integration of shadow into the ego to describe the first part of the journey.
• Integral Paradigm 101 14 January 2010 19:22 UTC www.audettesophia.com [Source type: FILTERED WITH BAYES]

^ The formal definition of a definite integral is stated in terms of the limit of a Riemann sum.
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

.Although all bounded piecewise continuous functions are Riemann integrable on a bounded interval, subsequently more general functions were considered, to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral, founded in measure theory (a subfield of real analysis).^ The theory of the Riemann integral was not fully satisfactory.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ In this discussion we consider only integrals of functions from [a,b] to R .
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ Evaluate the definite integral of the symmetric function .
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

.Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed.^ As indicated by the Venn diagram above, not every Lebesgue integral can be viewed as a Riemann integral, or even as an improper Riemann integral.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ In his effort to explore the basic concepts of integral psychology with a minimum of metaphysical assumptions, Chaudhuri (1973a) proposed a number of “principal tenets” that form the basis for his approach to integral psychology.
• Integral Paradigm 101 14 January 2010 19:22 UTC www.audettesophia.com [Source type: FILTERED WITH BAYES]

^ Many important functions do not have a Riemann integral -- even after we extend the class of integrable functions slightly by allowing "improper" Riemann integrals.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

### Notation

Isaac Newton used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with $\dot{x}$ or $x'\,\!$, which Newton used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.
The modern notation for the indefinite integral was introduced by Gottfried Leibniz in 1675 (Burton 1988, p. 359; Leibniz 1899, p. 154). He adapted the integral symbol, , from an elongated letter s, standing for summa (Latin for "sum" or "total"). .The modern notation for the definite integral, with limits above and below the integral sign, was first used by Joseph Fourier in Mémoires of the French Academy around 1819–20, reprinted in his book of 1822 (Cajori 1929, pp. 249–250; Fourier 1822, §231).^ If f is continuous on [a,b], the definite integral with integrand f(x) and limits a and b is simply equal to the value of the antiderivative F(x) at b minus the value of F at a.
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

^ Evaluate the definite integral using the substitution rule .
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

^ To put the gauge integral into proper perspective, we should first review the definition of the Riemann integral.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

## Terminology and notation

.If a function has an integral, it is said to be integrable.^ Now, when we turn to the gauge integral or the Lebesgue integral, more functions are integrable, and so it is even harder to produce examples of non-integrable functions.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ Many important functions do not have a Riemann integral -- even after we extend the class of integrable functions slightly by allowing "improper" Riemann integrals.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ When we study any sort of integral, we prove that certain classes of functions are integrable, and we give examples.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

The function for which the integral is calculated is called the integrand. The region over which a function is being integrated is called the domain of integration. Usually this domain will be an interval in which case it is enough to give the limits of that interval, which are called the limits of integration. .If the integral does not have a domain of integration, it is considered indefinite (one with a domain is considered definite).^ Evaluate the definite integral by integration by parts .
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

^ With our current knowledge of integration, we can't find the general equation of this indefinite integral.
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

^ Find the general indefinite integral by integration by parts 10 .
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

.In general, the integrand may be a function of more than one variable, and the domain of integration may be an area, volume, a higher dimensional region, or even an abstract space that does not have a geometric structure in any usual sense.^ The n-dimensional Euclidean space RN, real valued functions on RN: limits, continuity and differentiability, the chain rule and the directional derivative, the gradient and its properties, implicit functions and inverse mappings, extremal problems and Lagrange multipliers, multiple integration: definition, applications and techniques, the Jacobian and change of variables.

^ It can be generalized to functions defined on the whole real line, or to functions defined on finite-dimensional spaces, but its definition and theory then become slightly more complicated.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ Now, when we turn to the gauge integral or the Lebesgue integral, more functions are integrable, and so it is even harder to produce examples of non-integrable functions.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

The simplest case, the integral of a real-valued function f of one real variable x on the interval [a, b], is denoted by
$\int_a^b f(x)\,dx .$
.The ∫ sign represents integration; a and b are the lower limit and upper limit, respectively, of integration, defining the domain of integration; f is the integrand, to be evaluated as x varies over the interval [a,b]; and dx is the variable of integration.^ Real functions of one variable: limits, continuity, continuity on a closed interval, monotonic functions and inverse functions.

In correct mathematical typography, the dx is separated from the integrand by a space (as shown). Some authors use an upright d (that is, dx instead of dx).
.The variable of integration dx has different interpretations depending on the theory being used.^ By changing x to a function with a different variable we are essentially using the The Substitution Rule in reverse.
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

.For example, it can be seen as strictly a notation indicating that x is a dummy variable of integration, as a reflection of the weights in the Riemann sum, a measure (in Lebesgue integration and its extensions), an infinitesimal (in non-standard analysis) or as an independent mathematical quantity: a differential form.^ As indicated by the Venn diagram above, not every Lebesgue integral can be viewed as a Riemann integral, or even as an improper Riemann integral.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ Of course, one still has to do a fair amount of work to prove that the Lebesgue measurable sets form a sigma-algebra and the Lebesgue measure is countably additive on those sets.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ Offered: W. MATH 426 Fundamental Concepts of Analysis (3) NW Lebesgue measure on the reals.

.More complicated cases may vary the notation slightly.^ It can be generalized to functions defined on the whole real line, or to functions defined on finite-dimensional spaces, but its definition and theory then become slightly more complicated.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ With slightly more complicated definitions, we could allow plus and minus infinity as possible values for the integrals.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ Henstock and Kurzweil have also considered replacing the expression f(s i )(t i t i1 ) with a more general expression g(s i , [t i1 ,t i ]), where g is a function of a real variable and an interval variable; but the results in that case are more complicated and will not be considered here.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

.In so-called modern Arabic mathematical notation, which aims at pre-university levels of education in the Arab world and is written from right to left, an inverted integral symbol is used (W3C 2006).^ The legal and philosophical formalism in which Leibniz had been trained allowed him to create his own symbolic system, including not just the integral sign but the same notation of differentials we still use.
• Oct. 29, 1675: Leibniz ∫ums It All Up, Seriesly | This Day In Tech | Wired.com 14 January 2010 19:22 UTC www.wired.com [Source type: General]

^ Here in the USA, we usually teach analysis on three levels: The freshman calculus course, which uses the Riemann integral and omits most proofs.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ I asked if you have evidence related to the uptake or level of adherence to various schools of integral thought in the real world.
• What does the Integral Movement represent? « Open Integral 14 January 2010 19:22 UTC www.openintegral.net [Source type: Original source]

## Introduction

.Integrals appear in many practical situations.^ But bear in mind that the practical side was present in the Aurobindonian tradition,.and in many other Integral movements, from the very beginning.
• What does the Integral Movement represent? « Open Integral 14 January 2010 19:22 UTC www.openintegral.net [Source type: Original source]

Consider a swimming pool. .If it is rectangular, then from its length, width, and depth we can easily determine the volume of water it can contain (to fill it), the area of its surface (to cover it), and the length of its edge (to rope it).^ You'll find the more common units of length, volume, and area as well as less familiar energy/work, pressure, and power units.
• Dr. Alice Christie's Technology Integration: Math 15 September 2009 16:20 UTC www.west.asu.edu [Source type: FILTERED WITH BAYES]

.But if it is oval with a rounded bottom, all of these quantities call for integrals.^ These do not represent all possible Integral/GCN transmissions only those we choose to receive .

^ These groups have different visions about what integral is all about and those visions should be spelled out; but ARINA and Polysemy need not agree with each other.
• What does the Integral Movement represent? « Open Integral 14 January 2010 19:22 UTC www.openintegral.net [Source type: Original source]

Practical approximations may suffice for such trivial examples, but precision engineering (of any discipline) requires exact and rigorous values for these elements.
Approximations to integral of √x from 0 to 1, with  5 right samples (above) and  12 left samples (below)
To start off, consider the curve y = f(x) between x = 0 and x = 1, with f(x) = √x. We ask:
What is the area under the function f, in the interval from 0 to 1?
and call this (yet unknown) area the integral of f. The notation for this integral will be
$\int_0^1 \sqrt x \, dx \,\!.$
.As a first approximation, look at the unit square given by the sides x = 0 to x = 1 and y = f(0) = 0 and y = f(1) = 1. Its area is exactly 1. As it is, the true value of the integral must be somewhat less.^ The pyramid is completed and we – each of us – must look at the pyramid we have constructed and ask ourselves how it exemplifies our transcendence, our true Self.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

^ Look at it this way, if you are self-contained, if the Sovereign Integral is indeed within you at all times, then where exactly do you need to ascend?
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

^ Estimating perimeter and area of irregular shapes using unit squares and grid paper • Estimating area using unit squares .

Decreasing the width of the approximation rectangles shall give a better result; so cross the interval in five steps, using the approximation points 0, 15, 25, and so on to 1. Fit a box for each step using the right end height of each curve piece, thus √15, √25, and so on to √1 = 1. Summing the areas of these rectangles, we get a better approximation for the sought integral, namely
$extstyle \sqrt {\frac {1} {5}} \left ( \frac {1} {5} - 0 \right ) + \sqrt {\frac {2} {5}} \left ( \frac {2} {5} - \frac {1} {5} \right ) + \cdots + \sqrt {\frac {5} {5}} \left ( \frac {5} {5} - \frac {4} {5} \right ) \approx 0.7497.\,\!$
.Notice that we are taking a sum of finitely many function values of f, multiplied with the differences of two subsequent approximation points.^ Offered: W. MATH 466 Numerical Analysis III (3) NW Basic principles of numerical analysis, classical interpolation and approximation formulas, finite differences and difference equations.

^ Multivariable differential calculus: mean value theorem, inverse and implicit function theorems, and Lagrange multipliers.

^ More precisely: Assume F is continuous, and that G is some function defined at every point of [a,b], and that F'(t) exists and equals G(t) for all but at most countably many values of t.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

We can easily see that the approximation is still too large. .Using more steps produces a closer approximation, but will never be exact: replacing the 5 subintervals by twelve as depicted, we will get an approximate value for the area of 0.6203, which is too small.^ Increase the number of rectangular prisms to generate volume approximations closer to the true value.
• Double Integral for Volume - Wolfram Demonstrations Project 14 January 2010 19:22 UTC demonstrations.wolfram.com [Source type: FILTERED WITH BAYES]

.The key idea is the transition from adding finitely many differences of approximation points multiplied by their respective function values to using infinitely fine, or infinitesimal steps.^ Offered: W. MATH 466 Numerical Analysis III (3) NW Basic principles of numerical analysis, classical interpolation and approximation formulas, finite differences and difference equations.

^ Estimating sums and differences of whole numbers by using appropriate strategies such as rounding, front-end estimation, and compatible numbers • Adding and subtracting decimals and money amounts .

^ Multivariable differential calculus: mean value theorem, inverse and implicit function theorems, and Lagrange multipliers.

.As for the actual calculation of integrals, the fundamental theorem of calculus, due to Newton and Leibniz, is the fundamental link between the operations of differentiating and integrating.^ All of Differential Calculus, some Integral Calculus .
• Comprehensive Ratings and Review of Calculus web sites 29 September 2009 17:46 UTC www.csam.montclair.edu [Source type: General]

^ Differential, Integral and other calculus topics .
• Comprehensive Ratings and Review of Calculus web sites 29 September 2009 17:46 UTC www.csam.montclair.edu [Source type: General]

^ The Fundamental Theorem of Calculus defines the relationship between the processes of differentiation and integration.
• Integrals Tutorial 12 September 2009 11:29 UTC www.nipissingu.ca [Source type: Reference]

.Applied to the square root curve, f(x) = x1/2, it says to look at the antiderivative F(x) = 23x3/2, and simply take F(1) − F(0), where 0 and 1 are the boundaries of the interval [0,1].^ Look at it this way, and I am not picking on Eastern spirituality, but you asked the question about kundalini and so I’ll respond accordingly, but what I am saying applies to all of the traditional methods and mental models regardless of their cultural roots.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

So the exact value of the area under the curve is computed formally as
$\int_0^1 \sqrt x \,dx = \int_0^1 x^{\frac{1}{2}} \,dx = F(1)- F(0) = { extstyle \frac 2 3}.$
(This is a case of a general rule, that for f(x) = xq, with q ≠ −1, the related function, the so-called antiderivative is F(x) = (xq+1)/(q + 1).)
The notation
$\int f(x) \, dx \,\!$
conceives the integral as a weighted sum, denoted by the elongated .s, of function values, f(x), multiplied by infinitesimal step widths, the so-called differentials, denoted by dx.^ Multivariable differential calculus: mean value theorem, inverse and implicit function theorems, and Lagrange multipliers.

^ Let F be a real-valued, differentiable function on [a,b].
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

The multiplication sign is usually omitted.
.Historically, after the failure of early efforts to rigorously interpret infinitesimals, Riemann formally defined integrals as a limit of weighted sums, so that the dx suggested the limit of a difference (namely, the interval width).^ In fact, the Henstock-Kurzweil formulation -- the gauge integral -- is considerably simpler than the Lebesgue idea, and its definition is only slightly different from the definition of the Riemann integral.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ Again, there are several different, equivalent ways to define the gauge integral; we shall give only the formulation which emphasizes the similarity between the Riemann and gauge integrals.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ There are several different equivalent ways to define the Riemann integral; among them, this one is best suited for our present purposes: .
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

.Shortcomings of Riemann's dependence on intervals and continuity motivated newer definitions, especially the Lebesgue integral, which is founded on an ability to extend the idea of "measure" in much more flexible ways.^ As indicated by the Venn diagram above, not every Lebesgue integral can be viewed as a Riemann integral, or even as an improper Riemann integral.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ (Perhaps we should deemphasize the other measures in the first graduate course on integration theory, in order to make the gauge integral fit into the course more readily.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ Now, when we turn to the gauge integral or the Lebesgue integral, more functions are integrable, and so it is even harder to produce examples of non-integrable functions.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

Thus the notation
$\int_A f(x) \, d\mu \,\!$
refers to a weighted sum in which the function values are partitioned, with μ measuring the weight to be assigned to each value. Here A denotes the region of integration.
.Differential geometry, with its "calculus on manifolds", gives the familiar notation yet another interpretation.^ Introduction to Taylor polynomials and Taylor series, vector geometry in three dimensions,introduction to multivariable differential calculus, double integrals in Cartesian and polar coordinates.

Now f(x) and dx become a differential form, ω = f(x) dx, a new differential operator d, known as the exterior derivative appears, and the fundamental theorem becomes the more general Stokes' theorem,
$\int_{A} \bold{d} \omega = \int_{\part A} \omega , \,\!$
from which .Green's theorem, the divergence theorem, and the fundamental theorem of calculus follow.^ Antiderivatives and the fundamental theorem of calculus with applications.

^ The gauge integral simplifies and strengthens the Fundamental Theorems of Calculus: .
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ Culminates in the theorems of Green and Stokes, along with the Divergence Theorem.

More recently, infinitesimals have reappeared with rigor, through modern innovations such as non-standard analysis. .Not only do these methods vindicate the intuitions of the pioneers; they also lead to new mathematics.^ Because these “bridges” express with the tools of the HMS, even when they try to reveal the “deeper objects in the room,” they are only able to reveal the general shape and outline.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

.Although there are differences between these conceptions of integral, there is considerable overlap.^ In fact, the Henstock-Kurzweil formulation -- the gauge integral -- is considerably simpler than the Lebesgue idea, and its definition is only slightly different from the definition of the Riemann integral.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ Again, there are several different, equivalent ways to define the gauge integral; we shall give only the formulation which emphasizes the similarity between the Riemann and gauge integrals.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ There are several different equivalent ways to define the Riemann integral; among them, this one is best suited for our present purposes: .
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

.Thus, the area of the surface of the oval swimming pool can be handled as a geometric ellipse, a sum of infinitesimals, a Riemann integral, a Lebesgue integral, or as a manifold with a differential form.^ As indicated by the Venn diagram above, not every Lebesgue integral can be viewed as a Riemann integral, or even as an improper Riemann integral.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ As indicated by the Venn diagram above, not every improper Riemann integral is a Lebesgue integral.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

^ In fact, the Henstock-Kurzweil formulation -- the gauge integral -- is considerably simpler than the Lebesgue idea, and its definition is only slightly different from the definition of the Riemann integral.
• An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]

The calculated result will be the same for all.

## Formal definitions

.There are many ways of formally defining an integral, not all of which are equivalent.^ There is no year or specific time that will define the era of transparency and expansion, the rise of the Sovereign Integral.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

^ There are many nuances to the Quantum Pause technique, and I would encourage you to discover them on your own, in your own way.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

^ Perspectives on Beauty: Of course, there is much more to the Integral Art experience than just sitting back and taking it all in.
• Integral Life 14 January 2010 19:22 UTC integrallife.com [Source type: General]

The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but also occasionally for pedagogical reasons. The most commonly used definitions of integral are Riemann integrals and Lebesgue integrals.

### Riemann integral

Integral approached as Riemann sum based on tagged partition, with irregular sampling positions and widths (max in red). True value is 3.76; estimate is 3.648.
.The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval.^ Power series, improper integrals, uniform continuity, fundamental theorems on continuous functions, theory of the Riemann integral.

^ Continuity and differentiability theorems for functions defined by integrals.

Let [a,b] be a closed interval of the real line; then a tagged partition of [a,b] is a finite sequence
$a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_{n-1} \le t_n \le x_n = b . \,\!$
Riemann sums converging as intervals halve, whether sampled at  right,  minimum,  maximum, or  left.
This partitions the interval [a,b] into n sub-intervals [xi−1, xi] indexed by i, each of which is "tagged" with a distinguished point ti ∈ [xi−1, xi]. A Riemann sum of a function f with respect to such a tagged partition is defined as
$\sum_{i=1}^{n} f(t_i) \Delta_i ;$
thus each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width. Let Δi = xixi−1 be the width of sub-interval i; then the mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, maxi=1…n Δi. The Riemann integral of a function f over the interval [a,b] is equal to S if:
For all ε > 0 there exists δ > 0 such that, for any tagged partition [a,b] with mesh less than δ, we have
$\left| S - \sum_{i=1}^{n} f(t_i)\Delta_i \right| < \epsilon.$
.When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum becomes an upper (respectively, lower) Darboux sum, suggesting the close connection between the Riemann integral and the Darboux integral.^ Through the interweaving of mathematical concepts and processes, students learn to value mathematics, display confidence in their mathematical ability, solve problems, and make connections between mathematics and other subjects.

### Lebesgue integral

.The Riemann integral is not defined for a wide range of functions and situations of importance in applications (and of interest in theory).^ Power series, improper integrals, uniform continuity, fundamental theorems on continuous functions, theory of the Riemann integral.

^ Continuity and differentiability theorems for functions defined by integrals.

^ Forgiveness is the active formula of self-assessment of your present situation and the application of new behaviors that are in resonance with the Sovereign Integral.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

For example, the Riemann integral can easily integrate density to find the mass of a steel beam, but cannot accommodate a steel ball resting on it. This motivates other definitions, under which a broader assortment of functions is integrable (Rudin 1987). The Lebesgue integral, in particular, achieves great flexibility by directing attention to the weights in the weighted sum.
The definition of the Lebesgue integral thus begins with a measure, μ. In the simplest case, the Lebesgue measure μ(A) of an interval A = [a,b] is its width, ba, so that the Lebesgue integral agrees with the (proper) Riemann integral when both exist. In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals.
To exploit this flexibility, Lebesgue integrals reverse the approach to the weighted sum. As Folland (1984, p. 56) puts it, "To compute the Riemann integral of f, one partitions the domain [a,b] into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of f".
One common approach first defines the integral of the indicator function of a measurable set A by:
$\int 1_A d\mu = \mu(A)$.
This extends by linearity to a measurable simple function s, which attains only a finite number, n, of distinct non-negative values:
\begin{align} \int s \, d\mu &{}= \int\left(\sum_{i=1}^{n} a_i 1_{A_i}\right) d\mu \ &{}= \sum_{i=1}^{n} a_i\int 1_{A_i} \, d\mu \ &{}= \sum_{i=1}^{n} a_i \, \mu(A_i) \end{align}
(where the image of Ai under the simple function s is the constant value ai). Thus if E is a measurable set one defines
$\int_E s \, d\mu = \sum_{i=1}^{n} a_i \, \mu(A_i \cap E) .$
Then for any non-negative measurable function f one defines
$\int_E f \, d\mu = \sup\left\{\int_E s \, d\mu\, \colon 0 \leq s\leq f ext{ and } s ext{ is a simple function}\right\};$
that is, the integral of f is set to be the supremum of all the integrals of simple functions that are less than or equal to f. A general measurable function f, is split into its positive and negative values by defining
\begin{align} f^+(x) &{}= \begin{cases} f(x), & ext{if } f(x) > 0 \ 0, & ext{otherwise} \end{cases} \ f^-(x) &{}= \begin{cases} -f(x), & ext{if } f(x) < 0 \ 0, & ext{otherwise} \end{cases} \end{align}
Finally, f is Lebesgue integrable if
$\int_E |f| \, d\mu < \infty , \,\!$
and then the integral is defined by
$\int_E f \, d\mu = \int_E f^+ \, d\mu - \int_E f^- \, d\mu . \,\!$
.When the measure space on which the functions are defined is also a locally compact topological space (as is the case with the real numbers R), measures compatible with the topology in a suitable sense (Radon measures, of which the Lebesgue measure is an example) and integral with respect to them can be defined differently, starting from the integrals of continuous functions with compact support.^ Continuity and differentiability theorems for functions defined by integrals.

^ MATH 575 Fundamental Concepts of Analysis (3) Hoffman, Toro Sets, real numbers, topology of metric spaces, normed linear spaces, multivariable calculus from an advanced viewpoint.

^ Offered: W. MATH 426 Fundamental Concepts of Analysis (3) NW Lebesgue measure on the reals.

.More precisely, the compactly supported functions form a vector space that carries a natural topology, and a (Radon) measure can be defined as any continuous linear functional on this space; the value of a measure at a compactly supported function is then also by definition the integral of the function.^ Continuity and differentiability theorems for functions defined by integrals.

^ If you are new to the integral aesthetic experience and are looking for a basic introduction to the major forms and functions of Integral Art, you will not want to miss this talk....
• Integral Life 14 January 2010 19:22 UTC integrallife.com [Source type: General]

^ Power series, improper integrals, uniform continuity, fundamental theorems on continuous functions, theory of the Riemann integral.

.One then proceeds to expand the measure (the integral) to more general functions by continuity, and defines the measure of a set as the integral of its indicator function.^ Continuity and differentiability theorems for functions defined by integrals.

^ Power series, improper integrals, uniform continuity, fundamental theorems on continuous functions, theory of the Riemann integral.

^ MATH 524 Real Analysis (5) First quarter of a three-quarter sequence covering the theory of measure and integration, point set topology, Banach spaces, Lp spaces, applications to the theory of functions of one and several real variables.

This is the approach taken by Bourbaki (2004) and a certain number of other authors. For details see Radon measures.

### Other integrals

Although the Riemann and Lebesgue integrals are the most widely used definitions of the integral, a number of others exist, including:

## Properties

### Linearity

• The collection of Riemann integrable functions on a closed interval [a, b] forms a vector space under the operations of pointwise addition and multiplication by a scalar, and the operation of integration
$f \mapsto \int_a^b f \; dx$
is a linear functional on this vector space. Thus, firstly, the collection of integrable functions is closed under taking linear combinations; and, secondly, the integral of a linear combination is the linear combination of the integrals,
$\int_a^b (\alpha f + \beta g)(x) \, dx = \alpha \int_a^b f(x) \,dx + \beta \int_a^b g(x) \, dx. \,$
• Similarly, the set of real-valued Lebesgue integrable functions on a given measure space E with measure μ is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral
$f\mapsto \int_E f d\mu$
is a linear functional on this vector space, so that
$\int_E (\alpha f + \beta g) \, d\mu = \alpha \int_E f \, d\mu + \beta \int_E g \, d\mu.$
.
• More generally, consider the vector space of all measurable functions on a measure space (E,μ), taking values in a locally compact complete topological vector space V over a locally compact topological field K, f : EV.^ MATH 544 Topology and Geometry of Manifolds (5) First quarter of a three-quarter sequence covering general topology, the fundamental group, covering spaces, topological and differentiable manifolds, vector fields, flows, the Frobenius theorem, Lie groups, homogeneous spaces, tensor fields, differential forms, Stokes's theorem, deRham cohomology.

^ MATH 441 Topology (3) NW Metric and topological spaces, convergence, continuity, finite products, connectedness, and compactness.

Then one may define an abstract integration map assigning to each function f an element of V or the symbol ,
$f\mapsto\int_E f d\mu, \,$
that is compatible with linear combinations. In this situation the linearity holds for the subspace of functions whose integral is an element of V (i.e. "finite"). .The most important special cases arise when K is R, C, or a finite extension of the field Qp of p-adic numbers, and V is a finite-dimensional vector space over K, and when K=C and V is a complex Hilbert space.^ Vector spaces and degrees of extensions.

.Linearity, together with some natural continuity properties and normalisation for a certain class of "simple" functions, may be used to give an alternative definition of the integral.^ However, integral bridges with skew angles smaller than twenty degrees may be idealized using the proposed model.

^ It is the breath, enabled through Nature, that is life-giving to the human instrument, and it is the human instrument that is life-giving to the Sovereign Integral within the manifested physical reality of Earth.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

^ Normally, if adequate continuity is provided between the slab, girders and abutment using a proper reinforcement detailing, connection elements at abutment-deck joints are assumed as rigid.

.This is the approach of Daniell for the case of real-valued functions on a set X, generalized by Nicolas Bourbaki to functions with values in a locally compact topological vector space.^ MATH 524 Real Analysis (5) First quarter of a three-quarter sequence covering the theory of measure and integration, point set topology, Banach spaces, Lp spaces, applications to the theory of functions of one and several real variables.

^ MATH 441 Topology (3) NW Metric and topological spaces, convergence, continuity, finite products, connectedness, and compactness.

See (Hildebrandt 1953) for an axiomatic characterisation of the integral.

### Inequalities for integrals

.A number of general inequalities hold for Riemann-integrable functions defined on a closed and bounded interval [a, b] and can be generalized to other notions of integral (Lebesgue and Daniell).^ Power series, improper integrals, uniform continuity, fundamental theorems on continuous functions, theory of the Riemann integral.

^ Continuity and differentiability theorems for functions defined by integrals.

• Upper and lower bounds. An integrable function f on [a, b], is necessarily bounded on that interval. Thus there are real numbers m and M so that mf (x) ≤ M for all x in [a, b]. Since the lower and upper sums of f over [a, b] are therefore bounded by, respectively, m(ba) and M(ba), it follows that
$m(b - a) \leq \int_a^b f(x) \, dx \leq M(b - a).$
• Inequalities between functions. If f(x) ≤ g(x) for each x in [a, b] then each of the upper and lower sums of f is bounded above by the upper and lower sums, respectively, of g. Thus
$\int_a^b f(x) \, dx \leq \int_a^b g(x) \, dx.$
This is a generalization of the above inequalities, as M(ba) is the integral of the constant function with value M over [a, b].
• Subintervals. If [c, d] is a subinterval of [a, b] and f(x) is non-negative for all x, then
$\int_c^d f(x) \, dx \leq \int_a^b f(x) \, dx.$
$(fg)(x)= f(x) g(x), \; f^2 (x) = (f(x))^2, \; |f| (x) = |f(x)|.\,$
If f is Riemann-integrable on [a, b] then the same is true for |f|, and
$\left| \int_a^b f(x) \, dx \right| \leq \int_a^b | f(x) | \, dx.$
Moreover, if f and g are both Riemann-integrable then f 2, g 2, and fg are also Riemann-integrable, and
$\left( \int_a^b (fg)(x) \, dx \right)^2 \leq \left( \int_a^b f(x)^2 \, dx \right) \left( \int_a^b g(x)^2 \, dx \right).$
This inequality, known as the .Cauchy–Schwarz inequality, plays a prominent role in Hilbert space theory, where the left hand side is interpreted as the inner product of two square-integrable functions f and g on the interval [a, b].^ Review of Banach, Hilbert, and Lp spaces; locally convex spaces (duality and separation theory, distributions, and function spaces); operators on locally convex spaces (adjoints, closed graph/open mapping and Banach-Steinhaus theorems); Banach algebras (spectral theory, elementary applications); spectral theorem for Hilbert space operators.

• Hölder's inequality. Suppose that p and q are two real numbers, 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1, and f and g are two Riemann-integrable functions. Then the functions |f|p and |g|q are also integrable and the following Hölder's inequality holds:
$\left|\int f(x)g(x)\,dx\right| \leq \left(\int \left|f(x)\right|^p\,dx \right)^{1/p} \left(\int\left|g(x)\right|^q\,dx\right)^{1/q}.$
For p = q = 2, Hölder's inequality becomes the Cauchy–Schwarz inequality.
• Minkowski inequality. Suppose that p ≥ 1 is a real number and f and g are Riemann-integrable functions. Then |f|p, |g|p and |f + g|p are also Riemann integrable and the following Minkowski inequality holds:
$\left(\int \left|f(x)+g(x)\right|^p\,dx \right)^{1/p} \leq \left(\int \left|f(x)\right|^p\,dx \right)^{1/p} + \left(\int \left|g(x)\right|^p\,dx \right)^{1/p}.$
An analogue of this inequality for Lebesgue integral is used in construction of Lp spaces.

### Conventions

In this section f is a real-valued Riemann-integrable function. The integral
$\int_a^b f(x) \, dx$
over an interval [a, b] is defined if a < b. This means that the upper and lower sums of the function f are evaluated on a partition a = x0x1 ≤ . . . ≤ xn = b whose values xi are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating f within intervals [xi , xi +1] where an interval with a higher index lies to the right of one with a lower index. The values a and b, the end-points of the interval, are called the limits of integration of f. Integrals can also be defined if a > b:
• Reversing limits of integration. If a > b then define
$\int_a^b f(x) \, dx = - \int_b^a f(x) \, dx.$
This, with a = b, implies:
• Integrals over intervals of length zero. If a is a real number then
$\int_a^a f(x) \, dx = 0.$
.The first convention is necessary in consideration of taking integrals over subintervals of [a, b]; the second says that an integral taken over a degenerate interval, or a point, should be zero.^ Is not saying one is a Sovereign Integral and therefore First Source identical to saying one is God and enlightened or 'at one'?
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

One reason for the first convention is that the integrability of f on an interval [a, b] implies that f is integrable on any subinterval [c, d], but in particular integrals have the property that:
• Additivity of integration on intervals. If c is any element of [a, b], then
$\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx.$
With the first convention the resulting relation
\begin{align} \int_a^c f(x) \, dx &{}= \int_a^b f(x) \, dx - \int_c^b f(x) \, dx \ &{} = \int_a^b f(x) \, dx + \int_b^c f(x) \, dx \end{align}
is then well-defined for any cyclic permutation of a, b, and c.
Instead of viewing the above as conventions, one can also adopt the point of view that integration is performed of differential forms on oriented manifolds only. If M is such an oriented m-dimensional manifold, and M is the same manifold with opposed orientation and ω is an m-form, then one has:
$\int_M \omega = - \int_{M'} \omega \,.$
These conventions correspond to interpreting the integrand as a differential form, integrated over a chain. .In measure theory, by contrast, one interprets the integrand as a function f with respect to a measure μ, and integrates over a subset A, without any notion of orientation; one writes $extstyle{\int_A f\,d\mu = \int_{[a,b]} f\,d\mu}$ to indicate integration over a subset A. This is a minor distinction in one dimension, but becomes subtler on higher dimensional manifolds; see Differential form: Relation with measures for details.^ In knot theory, two embedded circles (knots) are considered equivalent if one may be smoothly deformed into the other without any cuts or self-intersections.
• Dr. Alice Christie's Technology Integration: Math 15 September 2009 16:20 UTC www.west.asu.edu [Source type: FILTERED WITH BAYES]

^ They love to see higher dimensions as if seeing is believing.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

^ MATH 524 Real Analysis (5) First quarter of a three-quarter sequence covering the theory of measure and integration, point set topology, Banach spaces, Lp spaces, applications to the theory of functions of one and several real variables.

## Fundamental theorem of calculus

.The fundamental theorem of calculus is the statement that differentiation and integration are inverse operations: if a continuous function is first integrated and then differentiated, the original function is retrieved.^ Antiderivatives and the fundamental theorem of calculus with applications.

^ Continuity and differentiability theorems for functions defined by integrals.

^ Offered: A. MATH 425 Fundamental Concepts of Analysis (3) NW One-variable differential calculus: chain rule, inverse function theorem, Rolle's theorem, intermediate value theorem, Taylor's theorem, and intermediate value theorem for derivatives.

.An important consequence, sometimes called the second fundamental theorem of calculus, allows one to compute integrals by using an antiderivative of the function to be integrated.^ Antiderivatives and the fundamental theorem of calculus with applications.

^ The second prophecy was the Grand Portal, and this has to do with not only one individual discovering the portal into the dimensions of the Sovereign Integral, but all of humanity.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

^ Offered: A. MATH 425 Fundamental Concepts of Analysis (3) NW One-variable differential calculus: chain rule, inverse function theorem, Rolle's theorem, intermediate value theorem, Taylor's theorem, and intermediate value theorem for derivatives.

### Statements of theorems

.
• Fundamental theorem of calculus. Let f be a real-valued integrable function defined on a closed interval [a, b].^ Antiderivatives and the fundamental theorem of calculus with applications.

^ Offered: A. MATH 425 Fundamental Concepts of Analysis (3) NW One-variable differential calculus: chain rule, inverse function theorem, Rolle's theorem, intermediate value theorem, Taylor's theorem, and intermediate value theorem for derivatives.

^ Multivariable differential calculus: mean value theorem, inverse and implicit function theorems, and Lagrange multipliers.

If F is defined for x in [a, b] by
$F(x) = \int_a^x f(t)\, dt.$
then F is continuous on [a, b]. If f is continuous at x in [a, b], then F is differentiable at x, and F ′(x) = f(x).
.
• Second fundamental theorem of calculus.^ Antiderivatives and the fundamental theorem of calculus with applications.

Let f be a real-valued integrable function defined on a closed interval [a, b]. If F is a function such that F ′(x) = f(x) for all x in [a, b] (that is, F is an antiderivative of f), then
$\int_a^b f(t)\, dt = F(b) - F(a).$
• Corollary. If f is a continuous function on [a, b], then f is integrable on [a, b], and F, defined by
$F(x) = \int_a^x f(t) \, dt$
is an anti-derivative of f on [a, b]. Moreover,
$\int_a^b f(t) \, dt = F(b) - F(a).$

## Extensions

### Improper integrals

The improper integral
$\int_{0}^{\infty} \frac{dx}{(x+1)\sqrt{x}} = \pi$
has unbounded intervals for both domain and range.
A "proper" Riemann integral assumes the integrand is defined and finite on a closed and bounded interval, bracketed by the limits of integration. An improper integral occurs when one or more of these conditions is not satisfied. In some cases such integrals may be defined by considering the limit of a sequence of proper Riemann integrals on progressively larger intervals.
If the interval is unbounded, for instance at its upper end, then the improper integral is the limit as that endpoint goes to infinity.
$\int_{a}^{\infty} f(x)dx = \lim_{b o \infty} \int_{a}^{b} f(x)dx$
If the integrand is only defined or finite on a half-open interval, for instance (a,b], then again a limit may provide a finite result.
$\int_{a}^{b} f(x)dx = \lim_{\epsilon o 0} \int_{a+\epsilon}^{b} f(x)dx$
.That is, the improper integral is the limit of proper integrals as one endpoint of the interval of integration approaches either a specified real number, or ∞, or −∞.^ It only means that in the deeper reality of the Sovereign Integral, the GSSC is revealed to be a form of suppression and it is not connected in any substantive way to either the Sovereign Integral or First Source.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

^ Realization of the Sovereign Integral consciousness is realization of one’s True Self as present in everyone else.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

In more complicated cases, limits are required at both endpoints, or at interior points.
Consider, for example, the function $frac{1}{(x+1)\sqrt{x}}$ integrated from 0 to ∞ (shown right). .At the lower bound, as x goes to 0 the function goes to ∞, and the upper bound is itself ∞, though the function goes to 0. Thus this is a doubly improper integral.^ Power series, improper integrals, uniform continuity, fundamental theorems on continuous functions, theory of the Riemann integral.

Integrated, say, from 1 to 3, an ordinary Riemann sum suffices to produce a result of $frac{\pi}{6}$. To integrate from 1 to ∞, a Riemann sum is not possible. However, any finite upper bound, say t (with t > 1), gives a well-defined result, $frac{\pi}{2} - 2\arctan frac{1}{\sqrt{t}}$. This has a finite limit as t goes to infinity, namely $frac{\pi}{2}$. Similarly, the integral from 13 to 1 allows a Riemann sum as well, coincidentally again producing $frac{\pi}{6}$. Replacing 13 by an arbitrary positive value s (with s < 1) is equally safe, giving $- frac{\pi}{2} + 2\arctan frac{1}{\sqrt{s}}$. This, too, has a finite limit as s goes to zero, namely $frac{\pi}{2}$. Combining the limits of the two fragments, the result of this improper integral is
\begin{align} \int_{0}^{\infty} \frac{dx}{(x+1)\sqrt{x}} &{} = \lim_{s o 0} \int_{s}^{1} \frac{dx}{(x+1)\sqrt{x}} + \lim_{t o \infty} \int_{1}^{t} \frac{dx}{(x+1)\sqrt{x}} \ &{} = \lim_{s o 0} \left( - \frac{\pi}{2} + 2 \arctan\frac{1}{\sqrt{s}} \right) + \lim_{t o \infty} \left( \frac{\pi}{2} - 2 \arctan\frac{1}{\sqrt{t}} \right) \ &{} = \frac{\pi}{2} + \frac{\pi}{2} \ &{} = \pi . \end{align}
This process does not guarantee success; a limit may fail to exist, or may be unbounded. For example, over the bounded interval 0 to 1 the integral of $frac{1}{x}$ does not converge; and over the unbounded interval 1 to ∞ the integral of $frac{1}{\sqrt{x}}$ does not converge.
.It may also happen that an integrand is unbounded at an interior point, in which case the integral must be split at that point, and the limit integrals on both sides must exist and must be bounded.^ My role, as single-pointed as it may seem, is to re-introduce the Sovereign Integral to humanity.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

Thus
\begin{align} \int_{-1}^{1} \frac{dx}{\sqrt[3]{x^2}} &{} = \lim_{s o 0} \int_{-1}^{-s} \frac{dx}{\sqrt[3]{x^2}} + \lim_{t o 0} \int_{t}^{1} \frac{dx}{\sqrt[3]{x^2}} \ &{} = \lim_{s o 0} 3(1-\sqrt[3]{s}) + \lim_{t o 0} 3(1-\sqrt[3]{t}) \ &{} = 3 + 3 \ &{} = 6. \end{align}
But the similar integral
$\int_{-1}^{1} \frac{dx}{x} \,\!$
cannot be assigned a value in this way, as the integrals above and below zero do not independently converge. (However, see Cauchy principal value.)

### Multiple integration

Double integral as volume under a surface.
.Integrals can be taken over regions other than intervals.^ The moment that the Sovereign Integral reaches into your HMS and announces its presence, you will never forget, nor will you mistake it for anything other than what it is.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

In general, an integral over a set E of a function f is written:
$\int_E f(x) \, dx.$
Here x need not be a real number, but can be another suitable quantity, for instance, a vector in R3. Fubini's theorem shows that such integrals can be rewritten as an iterated integral. .In other words, the integral can be calculated by integrating one coordinate at a time.^ One prophecy relates to the Grand Portal (as it is known within WingMakers) and the other to the End of Time.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

^ In other words, “moments”, in this definition, are passages of time or events.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

.Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function and the plane which contains its domain.^ MATH 324 Advanced Multivariable Calculus I (3) NW Topics include double and triple integrals, the chain rule, vector fields, line and surface integrals.

^ Continuity and differentiability theorems for functions defined by integrals.

^ There are numerous different races and just as in Nature, there is a natural selection process that determines which species will attain a dominant position among the various races.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

.(The same volume can be obtained via the triple integral — the integral of a function in three variables — of the constant function f(x, y, z) = 1 over the above mentioned region between the surface and the plane.^ MATH 324 Advanced Multivariable Calculus I (3) NW Topics include double and triple integrals, the chain rule, vector fields, line and surface integrals.

) .If the number of variables is higher, then the integral represents a hypervolume, a volume of a solid of more than three dimensions that cannot be graphed.^ The motivation was more subtle: it was to enslave the Sovereign Integral consciousness, knowing that it was more powerful, more intelligent, and more aware than even Anu.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

^ There is nothing more exotic or out of the ordinary than the language and dwellings of the Sovereign Integral, and the deeper you travel into this land, the stranger it will seem.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

^ Introduction to Taylor polynomials and Taylor series, vector geometry in three dimensions,introduction to multivariable differential calculus, double integrals in Cartesian and polar coordinates.

For example, the volume of the cuboid of sides 4 × 6 × 5 may be obtained in two ways:
• By the double integral
$\iint_D 5 \ dx\, dy$
of the function f(x, y) = 5 calculated in the region D in the xy-plane which is the base of the cuboid. For example, if a rectangular base of such a cuboid is given via the xy inequalities 2 ≤ x ≤ 7, 4 ≤ y ≤ 9, our above double integral now reads
$\int_4^9 \int_2^7 \ 5 \ dx\, dy$
From here, integration is conducted with respect to either x or y first; in this example, integration is first done with respect to x as the interval corresponding to x is the inner integral. Once the first integration is completed via the F(b) − F(a) method or otherwise, the result is again integrated with respect to the other variable. The result will equate to the volume under the surface.
• By the triple integral
$\iiint_\mathrm{cuboid} 1 \, dx\, dy\, dz$
of the constant function 1 calculated on the cuboid itself.

### Line integrals

A line integral sums together elements along a curve.
.The concept of an integral can be extended to more general domains of integration, such as curved lines and surfaces.^ MATH 324 Advanced Multivariable Calculus I (3) NW Topics include double and triple integrals, the chain rule, vector fields, line and surface integrals.

.Such integrals are known as line integrals and surface integrals respectively.^ MATH 324 Advanced Multivariable Calculus I (3) NW Topics include double and triple integrals, the chain rule, vector fields, line and surface integrals.

These have important applications in physics, as when dealing with vector fields.
A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve. Various different line integrals are in use. In the case of a closed curve it is also called a contour integral.
The function to be integrated may be a scalar field or a vector field. .The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve).^ MATH 324 Advanced Multivariable Calculus I (3) NW Topics include double and triple integrals, the chain rule, vector fields, line and surface integrals.

^ Introduction to Taylor polynomials and Taylor series, vector geometry in three dimensions,introduction to multivariable differential calculus, double integrals in Cartesian and polar coordinates.

This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulas in physics have natural continuous analogs in terms of line integrals; for example, the fact that work is equal to force, F, multiplied by displacement, s, may be expressed (in terms of vector quantities) as:
$W=\vec F\cdot\vec s.$
For an object moving along a path in a vector field $\vec F$ such as an electric field or gravitational field, the total work done by the field on the object is obtained by summing up the differential work done in moving from $\vec s$ to $\vec s + d\vec s$. This gives the line integral
$W=\int_C \vec F\cdot d\vec s.$

### Surface integrals

The definition of surface integral relies on splitting the surface into small surface elements.
.A surface integral is a definite integral taken over a surface (which may be a curved set in space); it can be thought of as the double integral analog of the line integral.^ MATH 324 Advanced Multivariable Calculus I (3) NW Topics include double and triple integrals, the chain rule, vector fields, line and surface integrals.

The function to be integrated may be a scalar field or a vector field. .The value of the surface integral is the sum of the field at all points on the surface.^ MATH 324 Advanced Multivariable Calculus I (3) NW Topics include double and triple integrals, the chain rule, vector fields, line and surface integrals.

This can be achieved by splitting the surface into surface elements, which provide the partitioning for Riemann sums.
.For an example of applications of surface integrals, consider a vector field v on a surface S; that is, for each point x in S, v(x) is a vector.^ MATH 324 Advanced Multivariable Calculus I (3) NW Topics include double and triple integrals, the chain rule, vector fields, line and surface integrals.

Imagine that we have a fluid flowing through S, such that v(x) determines the velocity of the fluid at x. The flux is defined as the quantity of fluid flowing through S in unit amount of time. To find the flux, we need to take the dot product of v with the unit surface normal to S at each point, which will give us a scalar field, which we integrate over the surface:
$\int_S {\mathbf v}\cdot \,d{\mathbf {S}}.$
The fluid flux in this example may be from a physical fluid such as water or air, or from electrical or magnetic flux. Thus surface integrals have applications in physics, particularly with the classical theory of electromagnetism.

### Integrals of differential forms

.A differential form is a mathematical concept in the fields of multivariable calculus, differential topology and tensors.^ MATH 575 Fundamental Concepts of Analysis (3) Hoffman, Toro Sets, real numbers, topology of metric spaces, normed linear spaces, multivariable calculus from an advanced viewpoint.

^ MATH 324 Advanced Multivariable Calculus I (3) NW Topics include double and triple integrals, the chain rule, vector fields, line and surface integrals.

^ MATH 544 Topology and Geometry of Manifolds (5) First quarter of a three-quarter sequence covering general topology, the fundamental group, covering spaces, topological and differentiable manifolds, vector fields, flows, the Frobenius theorem, Lie groups, homogeneous spaces, tensor fields, differential forms, Stokes's theorem, deRham cohomology.

The modern notation for the differential form, as well as the idea of the differential forms as being the wedge products of exterior derivatives forming an exterior algebra, was introduced by Élie Cartan.
We initially work in an open set in Rn. A 0-form is defined to be a smooth function f. When we integrate a function f over an m-dimensional subspace S of Rn, we write it as
$\int_S f\,dx^1 \cdots dx^m.$
(The superscripts are indices, not exponents.) We can consider dx1 through dxn to be formal objects themselves, rather than tags appended to make integrals look like Riemann sums. Alternatively, we can view them as covectors, and thus a measure of "density" (hence integrable in a general sense). We call the dx1, …,dxn basic 1-forms.
We define the wedge product, "∧", a bilinear "multiplication" operator on these elements, with the alternating property that
$dx^a \wedge dx^a = 0 \,\!$
for all indices a. Note that alternation along with linearity and associativity implies dxbdxa = −dxadxb. This also ensures that the result of the wedge product has an orientation.
We define the set of all these products to be basic 2-forms, and similarly we define the set of products of the form dxadxbdxc to be basic 3-forms. A general k-form is then a weighted sum of basic k-forms, where the weights are the smooth functions f. Together these form a vector space with basic k-forms as the basis vectors, and 0-forms (smooth functions) as the field of scalars. The wedge product then extends to k-forms in the natural way. Over Rn at most n covectors can be linearly independent, thus a k-form with k > n will always be zero, by the alternating property.
In addition to the wedge product, there is also the exterior derivative operator d. This operator maps k-forms to (k+1)-forms. For a k-form ω = f dxa over Rn, we define the action of d by:
${\bold d}{\omega} = \sum_{i=1}^n \frac{\partial f}{\partial x_i} dx^i \wedge dx^a.$
with extension to general k-forms occurring linearly.
This more general approach allows for a more natural coordinate-free approach to integration on manifolds. It also allows for a natural generalisation of the fundamental theorem of calculus, called Stokes' theorem, which we may state as
$\int_{\Omega} {\bold d}\omega = \int_{\partial\Omega} \omega \,\!$
where ω is a general k-form, and ∂Ω denotes the boundary of the region Ω. Thus, in the case that ω is a 0-form and Ω is a closed interval of the real line, this reduces to the fundamental theorem of calculus. In the case that ω is a 1-form and Ω is a two-dimensional region in the plane, the theorem reduces to Green's theorem. Similarly, using 2-forms, and 3-forms and Hodge duality, we can arrive at Stokes' theorem and the divergence theorem. In this way we can see that differential forms provide a powerful unifying view of integration.

## Methods

### Computing integrals

.The most basic technique for computing definite integrals of one real variable is based on the fundamental theorem of calculus.^ The realizations of the Sovereign Integral come in waves for most people, they are like layers being peeled, one at a time, that gradually allow the full realization, and when this realization may occur for each of you is a thing best left in the mystery.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

^ Realization of the Sovereign Integral consciousness is realization of one’s True Self as present in everyone else.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

Let f(x) be the function of x to be integrated over a given interval [a, b]. Then, find an antiderivative of f; that is, a function F such that F' = f on the interval. By the fundamental theorem of calculus—provided the integrand and integral have no singularities on the path of integration—$\int_a^b f(x)\,dx = F(b)-F(a).$
The integral is not actually the antiderivative, but the fundamental theorem provides a way to use antiderivatives to evaluate definite integrals.
The most difficult step is usually to find the antiderivative of f. It is rarely possible to glance at a function and write down its antiderivative. More often, it is necessary to use one of the many techniques that have been developed to evaluate integrals. .Most of these techniques rewrite one integral as a different one which is hopefully more tractable.^ It is these systems of control, “stacked” one on top of another, that ultimately slowed down the perception of time, and, in a sense, enabled Anu to operate in an entirely different time.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

Techniques include:
Alternate methods exist to compute more complex integrals. Many nonelementary integrals can be expanded in a Taylor series and integrated term by term. Occasionally, the resulting infinite series can be summed analytically. The method of convolution using Meijer G-functions can also be used, assuming that the integrand can be written as a product of Meijer G-functions. .There are also many less common ways of calculating definite integrals; for instance, Parseval's identity can be used to transform an integral over a rectangular region into an infinite sum.^ There are many nuances to the Quantum Pause technique, and I would encourage you to discover them on your own, in your own way.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

^ There is nothing more exotic or out of the ordinary than the language and dwellings of the Sovereign Integral, and the deeper you travel into this land, the stranger it will seem.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

^ In short, it is the transformation of the human instrument into a tool of expression for the Sovereign Integral state of consciousness.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

Occasionally, an integral can be evaluated by a trick; for an example of this, see Gaussian integral.
Computations of volumes of solids of revolution can usually be done with disk integration or shell integration.
Specific results which have been worked out by various techniques are collected in the list of integrals.

### Symbolic algorithms

Many problems in mathematics, physics, and engineering involve integration where an explicit formula for the integral is desired. Extensive tables of integrals have been compiled and published over the years for this purpose. .With the spread of computers, many professionals, educators, and students have turned to computer algebra systems that are specifically designed to perform difficult or tedious tasks, including integration.^ This is a difficult path for many people to accept, but this is what Quantum Pause is designed to achieve.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

Symbolic integration presents a special challenge in the development of such systems.
A major mathematical difficulty in symbolic integration is that in many cases, a closed formula for the antiderivative of a rather simple-looking function does not exist. .For instance, it is known that the antiderivatives of the functions exp ( x2), xx and sin x /x cannot be expressed in the closed form involving only rational and exponential functions, logarithm, trigonometric and inverse trigonometric functions, and the operations of multiplication and composition; in other words, none of the three given functions is integrable in elementary functions.^ The “wow” experience may manifest in a form that your HMS cannot interpret or translate into images, words, feelings, and thoughts.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

Differential Galois theory provides general criteria that allow one to determine whether the antiderivative of an elementary function is elementary. Unfortunately, it turns out that functions with closed expressions of antiderivatives are the exception rather than the rule. Consequently, computerized algebra systems have no hope of being able to find an antiderivative for a randomly constructed elementary function. On the positive side, if the 'building blocks' for antiderivatives are fixed in advance, it may be still be possible to decide whether the antiderivative of a given function can be expressed using these blocks and operations of multiplication and composition, and to find the symbolic answer whenever it exists. The Risch algorithm, implemented in Mathematica and other computer algebra systems, does just that for functions and antiderivatives built from rational functions, radicals, logarithm, and exponential functions.
Some special integrands occur often enough to warrant special study. In particular, it may be useful to have, in the set of antiderivatives, the special functions of physics (like the Legendre functions, the hypergeometric function, the Gamma function and so on). Extending the Risch-Norman algorithm so that it includes these functions is possible but challenging.
Most humans are not able to integrate such general formulae, so in a sense computers are more skilled at integrating highly complicated formulae. Very complex formulae are unlikely to have closed-form antiderivatives, so how much of an advantage this presents is a philosophical question that is open for debate.

.The integrals encountered in a basic calculus course are deliberately chosen for simplicity; those found in real applications are not always so accommodating.^ Those of us involved in the WingMakers and Lyricus are focused on introducing the Sovereign Integral state of consciousness and providing support to those interested in realizing this consciousness as their Self.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

.Some integrals cannot be found exactly, some require special functions which themselves are a challenge to compute, and others are so complex that finding the exact answer is too slow.^ Some refer to this as the soul, others refer to it as the astral body, but it is simply a sheath for the Sovereign Integral to operate within and it remains subject to the HMS and most of its programming.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

This motivates the study and application of numerical methods for approximating integrals, which today use floating-point arithmetic on digital electronic computers. Many of the ideas arose much earlier, for hand calculations; but the speed of general-purpose computers like the ENIAC created a need for improvements.
The goals of numerical integration are accuracy, reliability, efficiency, and generality. Sophisticated methods can vastly outperform a naive method by all four measures (Dahlquist & Björck 2008; Kahaner, Moler & Nash 1989; Stoer & Bulirsch 2002). Consider, for example, the integral
$\int_{-2}^{2} frac15 \left( frac{1}{100}(322 + 3 x (98 + x (37 + x))) - 24 \frac{x}{1+x^2} \right) dx ,$
which has the exact answer 9425 = 3.76. (In ordinary practice the answer is not known in advance, so an important task — not explored here — is to decide when an approximation is good enough.) A “calculus book” approach divides the integration range into, say, 16 equal pieces, and computes function values.
 x f(x) x f(x) −2.00 −1.50 −1.00 −0.50 0.00 0.50 1.00 1.50 2.00 2.22800 2.45663 2.67200 2.32475 0.64400 −0.92575 −0.94000 −0.16963 0.83600 −1.75 −1.25 −0.75 −0.25 0.25 0.75 1.25 1.75 2.33041 2.58562 2.62934 1.64019 −0.32444 −1.09159 −0.60387 0.31734
Numerical quadrature methods:  Rectangle,  Trapezoid,  Romberg,  Gauss
Using the left end of each piece, the rectangle method sums 16 function values and multiplies by the step width, h, here 0.25, to get an approximate value of 3.94325 for the integral. The accuracy is not impressive, but calculus formally uses pieces of infinitesimal width, so initially this may seem little cause for concern. Indeed, repeatedly doubling the number of steps eventually produces an approximation of 3.76001. However, 218 pieces are required, a great computational expense for such little accuracy; and a reach for greater accuracy can force steps so small that arithmetic precision becomes an obstacle.
A better approach replaces the horizontal tops of the rectangles with slanted tops touching the function at the ends of each piece. .This trapezium rule is almost as easy to calculate; it sums all 17 function values, but weights the first and last by one half, and again multiplies by the step width.^ Rather, it is an awareness of how we are all one and equal with First Source, and that this makes each of us responsible for the prison and the outflow of its dysfunctional events.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

^ And realization has, as its First Point, the unconditional oneness, equality and truthfulness of Self in all life expressions.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

^ Behind all of this, is First Source, who masterfully draws humanity to itself, one individual at a time.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

This immediately improves the approximation to 3.76925, which is noticeably more accurate. Furthermore, only 210 pieces are needed to achieve 3.76000, substantially less computation than the rectangle method for comparable accuracy.
Romberg's method builds on the trapezoid method to great effect. First, the step lengths are halved incrementally, giving trapezoid approximations denoted by T(h0), T(h1), and so on, where hk+1 is half of hk. .For each new step size, only half the new function values need to be computed; the others carry over from the previous size (as shown in the table above).^ Others focus only on messages of hope and light, and refuse to dwell on bad news.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

But the really powerful idea is to interpolate a polynomial through the approximations, and extrapolate to T(0). With this method a numerically exact answer here requires only four pieces (five function values)! The Lagrange polynomial interpolating {hk,T(hk)}k=0…2 = {(4.00,6.128), (2.00,4.352), (1.00,3.908)} is 3.76+0.148h2, producing the extrapolated value 3.76 at h = 0.
Gaussian quadrature often requires noticeably less work for superior accuracy. .In this example, it can compute the function values at just two x positions, ±2√3, then double each value and sum to get the numerically exact answer.^ There are numerous different races and just as in Nature, there is a natural selection process that determines which species will attain a dominant position among the various races.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

The explanation for this dramatic success lies in error analysis, and a little luck. An n-point Gaussian method is exact for polynomials of degree up to 2n−1. The function in this example is a degree 3 polynomial, plus a term that cancels because the chosen endpoints are symmetric around zero. (Cancellation also benefits the Romberg method.)
Shifting the range left a little, so the integral is from −2.25 to 1.75, removes the symmetry. Nevertheless, the trapezoid method is rather slow, the polynomial interpolation method of Romberg is acceptable, and the Gaussian method requires the least work — if the number of points is known in advance. As well, rational interpolation can use the same trapezoid evaluations as the Romberg method to greater effect.
Method Points Rel. Err. Value Trapezoid Romberg Rational Gauss 1048577 257 129 36 −5.3×10−13 −6.3×10−15 8.8×10−15 3.1×10−15 $extstyle \int_{-2.25}^{1.75} f(x)\,dx = 4.1639019006585897075\ldots$
In practice, each method must use extra evaluations to ensure an error bound on an unknown function; this tends to offset some of the advantage of the pure Gaussian method, and motivates the popular Gauss–Kronrod quadrature formulae. Symmetry can still be exploited by splitting this integral into two ranges, from −2.25 to −1.75 (no symmetry), and from −1.75 to 1.75 (symmetry). More broadly, adaptive quadrature partitions a range into pieces based on function properties, so that data points are concentrated where they are needed most.
Simpson's rule, named for Thomas Simpson (1710–1761), uses a parabolic curve to approximate integrals. In many cases, it is more accurate than the trapezoidal rule and others. The rule states that
$\int_{a}^{b} f(x) \, dx \approx \frac{b-a}{6}\left[f(a) + 4f\left(\frac{a+b}{2}\right)+f(b)\right],$
with an error of
$\left|-\frac{(b-a)^5}{2880} f^{(4)}(\xi)\right|.$
The computation of higher-dimensional integrals (for example, volume calculations) makes important use of such alternatives as Monte Carlo integration.
A calculus text is no substitute for numerical analysis, but the reverse is also true. .Even the best adaptive numerical code sometimes requires a user to help with the more demanding integrals.^ The motivation was more subtle: it was to enslave the Sovereign Integral consciousness, knowing that it was more powerful, more intelligent, and more aware than even Anu.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

.For example, improper integrals may require a change of variable or methods that can avoid infinite function values, and known properties like symmetry and periodicity may provide critical leverage.^ It is indisputably a part of the human instrument, and yet, the Sovereign Integral is constant, aware, awake, observant, alive, infinite, and, as odd as it may seem, it is not energy.
• Project Camelot | James: The Sovereign Integral 12 September 2009 11:29 UTC projectcamelot.org [Source type: Original source]

## Notes

1. ^ Shea, Marilyn (May 2007), Biography of Zu Chongzhi, University of Maine, retrieved 9 January 2009
Katz, Victor J. (2004), A History of Mathematics, Brief Version, Addison-Wesley, pp. 125–126, ISBN 978-0-321-16193-2
2. ^ Victor J. Katz (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3): 163-174 [165]
3. ^ Victor J. Katz (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3): 163–174 [165–9 & 173–4]
4. ^ http://www2.gol.com/users/coynerhm/0598rothman.html

# Wiktionary

Up to date as of January 15, 2010

## German

German Wikipedia has an article on:
Integralrechnung
Wikipedia de

### Noun

Integral n. (genitive Integrals, plural Integrale)
1. integral (notion in mathematics)

# Simple English

File:Integral as region under
Integration is about finding the surface s, given a, b and f(x)

An integral helps to find out how much space is under a graph of something. Integrals undo derivatives. A derivative helps to find the steepness of a graph.

This is the symbol for integration: $\int_\left\{\,\right\}^\left\{\,\right\}$ It was first used by Gottfried Wilhelm Leibniz who used it as a stylized s (for summa, Latin for sum).

Integrals and derivatives are part of a branch of mathematics called calculus. The links between these two is very important, and is called the Fundamental Theorem of Calculus[1].

Integration helps when trying to multiply units into a problem. For example, if a problem with rate ($distance \over time$) needs an answer with just distance, one solution is to integrate with respect to time. This means multiplying in time (to cancel the time in $distance \over time$). This is done by adding small slices of the rate graph together. The slices are close to zero in width, but adding them forever makes them add up to a whole. This is called a Riemann Sum.

Adding these slices together gives the equation that the first equation is the derivative of. Integrals are like a way to add many tiny things together by hand. It is like summation, which is adding $1+2+3+4....+n$. Integration is like we also have to add all the decimals and fractions in between as well.[1]

Another time integration is helpful is when finding the volume of a solid. It can add two-dimensional (without width) slices of the solid together forever until there is a width. This means the object now has three dimensions: the original two and a width. This gives the volume of the three-dimensional object described.

## Methods of Integration

### Antiderivative

By the fundamental theorem of calculus, the integral is the antiderivative.

If we take the function $2x$, for example, and anti-differentiate it, we can say that an integral of $2x$ is $x^2$. We say an integral, not the integral, because the antiderivative of a function is not unique. For example, $x^2+17$ also differentiates to $2x$. Because of this, when taking the antiderivative a constant C must be added. This is called an indefinite integral. This is because when finding the derivative of a function, constants equal 0, as in the function

$f\left(x\right) = 5x^2 + 9x + 15\,$.
$f\text{'}\left(x\right) = 10x + 9 + 0\,$. Note the 0: we cannot find it if we only have the derivative, so the integral is
$\int \left(10x + 9\right)\, dx = 5x^2 + 9x + C$.

### Simple Equations

A simple equation such as $y = x^2$ can be integrated with respect to x using the following technique. To integrate, you add 1 to the power x is raised to, and then divide x by the value of this new power. Therefore, integration of a normal equation follows this rule: $\int_\left\{\,\right\}^\left\{\,\right\} x^n dx = \frac\left\{x^\left\{n+1\right\}\right\}\left\{n+1\right\} + C$

The $dx$ at the end is what shows that we are integrating with respect to x, that is, as x changes. This can be seen to be the inverse of differentiation. However, there is a constant, C, added when you integrate. This is called the constant of integration. This is required because differentiating an integer results in zero, therefore integrating zero (which can be put onto the end of any integrand) produces an integer, C. The value of this integer would be found by using given conditions.

Equations with more than one terms are simply integrated by integrating each individual term:

$\int_\left\{\,\right\}^\left\{\,\right\} x^2 + 3x - 2 dx = \int_\left\{\,\right\}^\left\{\,\right\} x^2 dx + \int_\left\{\,\right\}^\left\{\,\right\} 3x dx - \int_\left\{\,\right\}^\left\{\,\right\} 2 dx = \frac\left\{x^3\right\}\left\{3\right\} + \frac\left\{3x^2\right\}\left\{2\right\} - 2x + C$

### Integration involving e and ln

There are certain rules for integrating using e and the natural logarithm. Most importantly, $e^x$ is the integral of itself (with the addition of a constant of integration): $\int_\left\{\,\right\}^\left\{\,\right\}e^\left\{x\right\} dx = e^\left\{x\right\} + C$

With e raised to a function of x ($f\left(x\right)$):

$\int_\left\{\,\right\}^\left\{\,\right\} e^\left\{f\left(x\right)\right\} dx = e^\left\{f\left(x\right)\right\}\div\int_\left\{\,\right\}^\left\{\,\right\}f\left(x\right) dx$

The natural logarithm, ln, is useful when integrating equations with $1/x$. These cannot be integrated using the formula above (add one to the power, divide by the power), because adding one to the power produces 0, and a division by 0 is not possible. Instead, the integral of $1/x$ is $\ln x$: $\int_\left\{\,\right\}^\left\{\,\right\}\frac\left\{1\right\}\left\{x\right\} dx = \ln x + C$

In a more general form: $\int_\left\{\,\right\}^\left\{\,\right\}\frac\left\{f\text{'}\left(x\right)\right\}\left\{f\left(x\right)\right\} dx = \ln \left\{|f\left(x\right)|\right\} + C$

The two vertical bars indicated a absolute value; the sign (positive or negative) of $f\left(x\right)$ is ignored. This is because there is no value for the natural logarithm of negative numbers.

## Properties

### Sum of functions

The integral of a sum of functions is the sum of each function's integral. that is,

$\int\limits_\left\{a\right\}^\left\{b\right\} \left[f\left(x\right) + g\left(x\right)\right]\, dx = \int\limits_\left\{a\right\}^\left\{b\right\} f\left(x\right)\, dx + \int\limits_\left\{a\right\}^\left\{b\right\} g\left(x\right)\, dx$.

Proof is straightforward: The definition of an integral is a limit of sums. Thus

$\int\limits_\left\{a\right\}^\left\{b\right\} \left[f\left(x\right) + g\left(x\right)\right]\, dx = \lim_\left\{n \to \infty\right\} \sum_\left\{i=1\right\}^n \left\left(f\left(x_i^*\right) + g\left(x_i^*\right)\right\right)$
$= \lim_\left\{n \to \infty\right\} \sum_\left\{i=1\right\}^n f\left(x_i^*\right) + \sum_\left\{i=1\right\}^n g\left(x_i^*\right)$
$= \lim_\left\{n \to \infty\right\} \sum_\left\{i=1\right\}^n f\left(x_i^*\right) + \lim_\left\{n \to \infty\right\} \sum_\left\{i=1\right\}^n g\left(x_i^*\right)$
$= \int\limits_\left\{a\right\}^\left\{b\right\} f\left(x\right)\, dx + \int\limits_\left\{a\right\}^\left\{b\right\} g\left(x\right)\, dx$

Note that both integrals have the same limits.

### Constants in integration

When a constant is in an integral with a function, the constant can be taken out. Further, when a constant c is not accompanied by a function, its value is c * x. That is,

$\int\limits_\left\{a\right\}^\left\{b\right\} cf\left(x\right)\, dx = c \int\limits_\left\{a\right\}^\left\{b\right\} f\left(x\right)\, dx$ and

This can only be done with a constant.

$\int\limits_\left\{a\right\}^\left\{b\right\} c\, dx = c\left(b-a\right)$

Proof is again by the definition of an integral.

### Other

If a, b and c are in order (i.e. after each other on the x-axis), the integral of f(x) from point a to point b plus the integral of f(x) from point b to c equals the integral from point a to c. That is,

$\int\limits_\left\{a\right\}^\left\{b\right\} f\left(x\right)\, dx + \int\limits_\left\{b\right\}^\left\{c\right\} f\left(x\right)\, dx = \int\limits_\left\{a\right\}^\left\{c\right\} f\left(x\right)\, dx$ if they are in order. (This also holds when a, b, c are not in order if we define $\int\limits_\left\{a\right\}^\left\{b\right\} f\left(x\right) \,dx= -\int\limits_\left\{b\right\}^\left\{a\right\} f\left(x\right)\, dx$.)
$\int\limits_\left\{a\right\}^\left\{a\right\} f\left(x\right)\, dx = 0$. This follows the fundamental theorem of calculus (FTC): F(a)-F(a)=0
$\int\limits_\left\{a\right\}^\left\{b\right\} f\left(x\right)\, dx = -\int\limits_\left\{b\right\}^\left\{a\right\} f\left(x\right)\, dx$ Again, following the FTC: $F\left(b\right)-F\left(a\right) = -\left[F\left(a\right)-F\left(b\right)\right]$

## References

1. 1.0 1.1 Barton, David; Stuart Laird (2003). "16". Delta Mathematics. Pearson Education. ISBN 0-582-54539-0.

# Citable sentences

Up to date as of December 13, 2010

Here are sentences from other pages on Integral, which are similar to those in the above article.