
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 $(2, \pi /3,5)$ in cylindrical coordinates. What is this point when expressed in $(x,y,z)$-coordinates?
Consider the point $(1, -1,5)$ in $(x,y,z)$-coordinates. What is this point when expressed in cylindrical coordinates where $0\le \theta <2\pi$?

Triple integrals in cylindrical coordinates

If you want to evaluate this integral you have to change $R$ to a region defined in $(x,y,z)$-coordinates, and change $\d V$ to some combination of $\d x\d y\d z$ leaving you with some iterated integral: Now consider representing a region $R$ in cylindrical coordinates and let’s express $\d V$ in terms of $\d r$, $\d \theta$, and $\d z$. 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 $a$ and height $h$.