PX275 - B1a - cylindrical polar coordinates

at point $P :$

\begin{matrix} x = ρ \cos ϕ \\ y = ρ \sin ϕ \\ ρ^{2} = x^{2} + y^{2} \end{matrix}

note:
$z \in (- \infty, \infty)$
$ϕ \in [0, 2 π)$
$ρ \in [0, \infty)$
it may sometimes be easier if the range of integration for $ϕ$ is in $ϕ \in [- π, π]$
for integration, the surface area and the volume elements need to be defined:

\begin{aligned} d S & = ρ d ϕ d z \\ d V & = ρ d ϕ d z d ρ \end{aligned}