Why does $dV=\rho^2\sin{\phi}\, d{\rho}\, d{\theta}\, d\phi$?

Question

I was watching this video from MichelvanBiezen wherein he explains how to find the volume of a hemisphere by setting $dV=\rho^2\sin{\phi}\, d{\rho}\, d{\theta}\, d\phi$. I understood everything in the video except for this first part, which he does not elaborate on. Could someone give an explanation or geometric intuition? Any help is greatly appreciated.

Answer 1

You have to visualize volume elements in spherical coordinates.

In Cartesian coordinates $dV$ is a small cube, which is why it's equal to $dxdydz$.

In spherical coordinates it's more complicated. But I'm pretty sure this picture will explain everything:

Source of the diagram

Note: Cartesian volume elements are not the same as spherical volume elements.