We integrate over regions in cylindrical coordinates.

The first way we will generalize polar coordinates to three dimensions is with cylindrical coordinates.
Consider the point in cylindrical coordinates. What is this point when expressed in -coordinates?
Consider the point in -coordinates. What is this point when expressed in cylindrical coordinates where ?

Triple integrals in cylindrical 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 cylindrical coordinates and let’s express in terms of , , and . To do this, consider the diagram below:

Here we see
Write down a triple integral in cylindrical coordinates that will compute the volume of a cylinder of radius and height .