cylindrical coordinates

Polar coordinates $(r,\theta)$ in the $xy$ -plane left unchanged, with $z$ added on; volume element $dV=r\,dz\,dr\,d\theta$ .

Cylindrical coordinates keep polar coordinates $(r,\theta)$ for the $xy$ -plane exactly as before and leave $z$ alone. The volume element $dV=r\,dz\,dr\,d\theta$ carries the same extra factor of $r$ a polar double integral already carries in $dA=r\,dr\,d\theta$ , since cylindrical coordinates don't touch $z$ . A solid is a good candidate whenever its description involves $x^2+y^2$ as a single block — a circle, disk, or cylinder — rather than $x$ and $y$ separately.

Examples

The volume under $z=4-x^2-y^2$ and above $z=0$ is $\int_0^{2\pi}\!\int_0^2\!\int_0^{4-r^2}r\,dz\,dr\,d\theta=8\pi$ .