In mathematics — in particular, in multivariable calculus — a volume integral refers to an integral over a 3dimensional domain.
Volume integral is a triple integral of the constant function 1, which gives the volume of the region D, that is, the integral
It can also mean a triple integral within a region D in R^{3} of a function f(x,y,z), and is usually written as:
A volume integral in cylindrical coordinates is
and a volume integral in spherical coordinates (using the standard convention for angles) has the form
Integrating the function f(x,y,z) = 1 over a unit cube yields the following result:
So the volume of the unit cube is 1 as expected. This is rather trivial however and a volume integral is far more powerful. For instance if we have a scalar function describing the density of the cube at a given point (x,y,z) by f = x + y + z then performing the volume integral will give the total mass of the cube:
Topics in Calculus  

Fundamental theorem Limits of functions Continuity Mean value theorem

In mathematics — in particular, in multivariable calculus — a volume integral refers to an integral over a 3dimensional domain.
Volume integral is a triple integral of the constant function 1, which gives the volume of the region D, that is, the integral
It can also mean a triple integral within a region D in R^{3} of a function $f(x,y,z),$ and is usually written as:
A volume integral in cylindrical coordinates is
and a volume integral in spherical coordinates (using the standard convention for angles) has the form
Integrating the function $f(x,y,z)\; =\; 1$ over a unit cube yields the following result:
$\backslash iiint\; \backslash limits\_0^1\; 1\; \backslash ,dx\backslash ,\; dy\; \backslash ,dz\; =\; \backslash iint\; \backslash limits\_0^1\; (1\; \; 0)\; \backslash ,dy\; \backslash ,dz\; =\; \backslash int\; \backslash limits\_0^1\; (1\; \; 0)\; dz\; =\; 1\; \; 0\; =\; 1$
So the volume of the unit cube is 1 as expected. This is rather trivial however and a volume integral is far more powerful. For instance if we have a scalar function $\backslash begin\{align\}\; f\backslash colon\; \backslash mathbb\{R\}^3\; \&\backslash to\; \backslash mathbb\{R\}\; \backslash end\{align\}$ describing the density of the cube at a given point $(x,y,z)$ by $f\; =\; x+y+z$ then performing the volume integral will give the total mass of the cube:
$\backslash iiint\; \backslash limits\_0^1\; \backslash left(x\; +\; y\; +\; z\backslash right)\; \backslash ,\; dx\; \backslash ,dy\; \backslash ,dz\; =\; \backslash iint\; \backslash limits\_0^1\; \backslash left(\backslash frac\; 12\; +\; y\; +\; z\backslash right)\; \backslash ,\; dy\; \backslash ,dz\; =\; \backslash int\; \backslash limits\_0^1\; \backslash left(1\; +\; z\backslash right)\; \backslash ,\; dz\; =\; \backslash frac\; 32$
