Having a spherical shell of internal radius of $r_0$ and thickness $\Delta r$, what is the volume element?
I started calculating and faced aome differential terms of degrees higher than 1. Should I ignore them? Would it be an approximation or this is the method to write elements?
The volume of the shell is $4\pi r_0^2 \Delta r$. You can think at a solid with base surface the sphere and ''height'' $\Delta r$.
Or you can calculate the volume as: $$ \frac{4}{3}\pi (r_0+\Delta r)^3-\frac{4}{3}\pi r_0^3 $$
and ignore the terms that are of second or higher order in $\Delta r$.
Or you can use the volume element $dV=r^2\sin \varphi d\theta d \varphi dr$ and integrate with respect to $\varphi$ form $0$ to $\pi$ and with respect to $\theta$ from $0$ to $2\pi$.