Let $G=\{ (x,y,z) : x^2+y^2+z^2 \leq 4 \text{ and } x^2+y^2 \geq 1\}. $
Use spherical coordinates to describe the area $G$.
I imagine that this describes the area in between the sphere with radius 4 and the cylinder of radius 1 but I don't know if that is correct or how to do that. Any hints?
I suggest you to use sperical coordinates.
You have to introduce three new variables $\rho,\psi$ and $\theta$ in order to get a new set $G\star$ to do the integration or, as your exercise, to give an other description of an area/volume. These variables are such that $(\rho,\theta,\psi)\in(0,+\infty)\times(0,\pi)\times(0,2\pi)$. Now we have to do this change of variables: \begin{cases} x=\rho \sin\theta\cos\psi \\ y=\rho \sin\theta\sin\psi \\ z=\rho\cos\theta \end{cases}
If you now substitute these new coordinates in your initial conditions that define your set $G$, you can easily find the new set $G\star$ which depens from $\rho,\theta$ and $\psi$.
If you have to integrate a volume or calculate the integral of a function in three variables, don't forget the absolute value of the determinat of the Jacobian matrix of the transformation $|\mathrm{det}(J_T)|=\rho^2\sin\theta$.
So, you have this relation between the "old" and the "new" differential: $\mathrm{d}x\mathrm{d}y\mathrm{d}z=|\mathrm{det}(J_T)|\mathrm{d}\rho\mathrm{d}\theta\mathrm{d}\psi=\rho^2\sin\theta\mathrm{d}\rho\mathrm{d}\theta\mathrm{d}\psi$.
If you have other difficulties let me know it.