We integrate over regions in spherical coordinates.

Another way to generalize polar coordinates to three dimensions is with spherical coordinates.
Consider the point in spherical coordinates. What is this point when expressed in -coordinates?

Triple integrals in spherical coordinates

If you want to evaluate this integral you have to change to a region defined in -coordinates, and change to some combination of leaving you with some iterated integral: Now consider representing a region in spherical coordinates and let’s express in terms of , , and . To do this, consider the diagram below:

Here we see

We may now state at theorem:

Write down a triple integral in spherical coordinates that will compute the volume of a sphere of radius .