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
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: