Volume integral


In mathematics—in particular, in multivariable calculus—a 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 of a function and is usually written as:
A volume integral in cylindrical coordinates is
and a volume integral in spherical coordinates has the form

Example 1

Integrating the equation 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 density function on the unit cube then the volume integral will give the total mass of the cube. For example for density function:
the total mass of the cube is: