Volume integral

In mathematicsin particular, in multivariable calculusa volume integral refers to an integral over a 3-dimensional domain, that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applications, for example, to calculate flux densities.

In coordinates

It can also mean a triple integral within a region D in R3 of a function and is usually written as:

A volume integral in cylindrical coordinates is

Volume element in spherical coordinates

and a volume integral in spherical coordinates (using the ISO convention for angles with as the azimuth and measured from the polar axis (see more on conventions)) has the form

Example 1

Integrating the function 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 by then performing the volume integral will give the total mass of the cube: