From Wikipedia, the free encyclopedia
A definite integral of a function can be represented as the signed area of the region bounded by its graph.
.^ 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 [
a,
b] of the
real line, the
definite integral
is defined informally to be the net signed
.^ 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]
.^ 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]
.^ 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]
.^ 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. integral@Everything2.com 14 January 2010 19:22 UTC everything2.com [Source type: FILTERED WITH BAYES]
^ 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]
.^ 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.
.^ In fact, the HenstockKurzweil 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. An Introduction to the Gauge Integral 14 January 2010 19:22 UTC www.math.vanderbilt.edu [Source type: FILTERED WITH BAYES]
.^ 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]
.^ 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 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 finitedimensional 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]
.^ Explore fourdimensional structures, projected into threedimensional space. Dr. Alice Christie's Technology Integration: Math 15 September 2009 16:20 UTC www.west.asu.edu [Source type: FILTERED WITH BAYES]
.^ Offered: A. MATH 442 Differential Geometry (3) NW Curves in 3space, continuity and differentiability in 3space, 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 .
.^ 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]
.^ 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
Precalculus 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 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]
.^ 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]
.^ 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 fatherandson mathematicians
Zu Chongzhi and
Zu Geng to find the volume of a sphere.
^{[1]} .^ 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]}
.^ 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.
.^ Problemsolving 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]
.^ 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]
.^ 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]} .^ 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, CalculusII . Comprehensive Ratings and Review of Calculus web sites 29 September 2009 17:46 UTC www.csam.montclair.edu [Source type: General]
.^ 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.
.^ 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]} .^ 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.
^ 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 PengYee has given a Riemanntype 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
.^ 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 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.
.^ 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 problemsolving situations with a greater understanding of the meanings of operations and how they relate to one another.
.^ 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.
.^ 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.
.^ 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 backandforth 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.
.^ Many use Jungian terminology to describe their growth process of integration of shadow into the ego to describe the first part of the journey.
^ 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]
.^ 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]
.^ 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.
^ 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
or
, 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").
.^ 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
.^ 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 nonintegrable 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.
.^ 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]
.^ The ndimensional 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 finitedimensional 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 nonintegrable 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 realvalued function f of one real variable x on the interval [a, b], is denoted by
.^ 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, d
x instead of
dx).
.^ 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]
.^ 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 sigmaalgebra 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.
.^ It can be generalized to functions defined on the whole real line, or to functions defined on finitedimensional 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 i1 ) with a more general expression g(s i , [t i1 ,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]
.^ 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
.^ 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.
.^ 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]
.^ 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
.^ 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 selfcontained, 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,
^{1}⁄
_{5},
^{2}⁄
_{5}, and so on to 1. Fit a box for each step using the right end height of each curve piece, thus √
^{1}⁄
_{5}, √
^{2}⁄
_{5}, and so on to √1 = 1. Summing the areas of these rectangles, we get a better approximation for the sought integral, namely
.^ 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.
.^ 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]
.^ 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, frontend 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.
.^ 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]
.^ 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
(This is a case of a general rule, that for
f(
x) =
x^{q}, with
q ≠ −1, the related function, the socalled
antiderivative is
F(
x) = (
x^{q+1})/(
q + 1).)
The notation
conceives the integral as a weighted sum, denoted by the elongated
.^ Multivariable differential calculus: mean value theorem, inverse and implicit function theorems, and Lagrange multipliers.
^ Let F be a realvalued, 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.
.^ In fact, the HenstockKurzweil 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]
.^ 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 nonintegrable 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
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.
.^ 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,
from which
.^ 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
nonstandard analysis.
.^ 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]
.^ In fact, the HenstockKurzweil 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]
.^ 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 HenstockKurzweil 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 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.
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.
.^ 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
Riemann sums converging as intervals halve, whether sampled at
■ right,
■ minimum,
■ maximum, or
■ left.
This partitions the interval [a,b] into n subintervals [x_{i−1}, x_{i}] indexed by i, each of which is "tagged" with a distinguished point t_{i} ∈ [x_{i−1}, x_{i}]. A Riemann sum of a function f with respect to such a tagged partition is defined as
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 subinterval, and width the same as the subinterval width. Let Δ_{i} = x_{i}−x_{i−1} be the width of subinterval i; then the mesh of such a tagged partition is the width of the largest subinterval formed by the partition, max_{i=1…n} Δ_{i}. 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
.^ 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
.^ 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 selfassessment 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,
b −
a, 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".
 .
This extends by linearity to a measurable
simple function s, which attains only a finite number,
n, of distinct nonnegative values:
(where the image of A_{i} under the simple function s is the constant value a_{i}). Thus if E is a measurable set one defines
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
Finally, f is Lebesgue integrable if
and then the integral is defined by
.^ 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.
.^ 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....
^ 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.
^ 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 threequarter 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:
 The RiemannStieltjes integral, an extension of the Riemann integral.
 The LebesgueStieltjes integral, further developed by Johann Radon, which generalizes the RiemannStieltjes and Lebesgue integrals.
 The Daniell integral, which subsumes the Lebesgue integral and LebesgueStieltjes integral without the dependence on measures.
 The HenstockKurzweil integral, variously defined by Arnaud Denjoy, Oskar Perron, and (most elegantly, as the gauge integral) Jaroslav Kurzweil, and developed by Ralph Henstock.
 The Itō integral and Stratonovich integral, which define integration with respect to stochastic processes such as Brownian motion.
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

 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,

 Similarly, the set of realvalued 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

 is a linear functional on this vector space, so that

.
 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 : E → V.^ MATH 544 Topology and Geometry of Manifolds (5) First quarter of a threequarter 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 ∞,

 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"). .^ Vector spaces and degrees of extensions.
.^ 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 lifegiving to the human instrument, and it is the human instrument that is lifegiving 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 abutmentdeck joints are assumed as rigid.
.^ MATH 524 Real Analysis (5) First quarter of a threequarter 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
.^ 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 m ≤ f (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(b − a) and M(b − a), it follows that

 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

 This is a generalization of the above inequalities, as M(b − a) 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 nonnegative for all x, then


 If f is Riemannintegrable on [a, b] then the same is true for f, and
 Moreover, if f and g are both Riemannintegrable then f ^{2}, g ^{2}, and fg are also Riemannintegrable, and
 This inequality, known as the .^ 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 BanachSteinhaus 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 Riemannintegrable functions. Then the functions f^{p} and g^{q} are also integrable and the following Hölder's inequality holds:
 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 Riemannintegrable functions. Then f^{p}, g^{p} and f + g^{p} are also Riemann integrable and the following Minkowski inequality holds:
 An analogue of this inequality for Lebesgue integral is used in construction of L^{p} spaces.
Conventions
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 =
x_{0} ≤
x_{1} ≤ . . . ≤
x_{n} =
b whose values
x_{i} are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating
f within intervals [
x_{ i} ,
x_{ i +1}] where an interval with a higher index lies to the right of one with a lower index. The values
a and
b, the endpoints 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

This, with a = b, implies:
 Integrals over intervals of length zero. If a is a real number then

.^ 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

With the first convention the resulting relation
is then welldefined 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
mdimensional manifold, and
M is the same manifold with opposed orientation and
ω is an
mform, then one has:
These conventions correspond to interpreting the integrand as a differential form, integrated over a
chain.
.^ In knot theory, two embedded circles (knots) are considered equivalent if one may be smoothly deformed into the other without any cuts or selfintersections. 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 threequarter 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
.^ 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 Onevariable differential calculus: chain rule, inverse function theorem, Rolle's theorem, intermediate value theorem, Taylor's theorem, and intermediate value theorem for derivatives.
.^ 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 Onevariable 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
.