I am trying to solve the following integral: $\int_{S_2} z^2 dS(x)$, where
$S_2=\{(x,y,z)\in \mathbb{R^3}$:$x^2+y^2+z^2=1$} and the Volume element $dS(x)=r^2sin\psi \quad dr d\phi d\psi $.
I parametrized $S_2$: $(0, \psi)$ $\in$ $(0, \pi)\times(0,2\pi)$ $\mapsto$ $(\sin \phi \cos\psi, \sin\phi \sin \psi, \cos \phi)$.
My idea: $\int_{S_2} z^2 dS(x)$ = $\int_{0}^{2\pi}$$\int_{0}^{\pi}$$\int_{0}^{1}$$\cos^2(\phi) r^2 sin \psi drd\phi\psi$.
I don't know what to do next... Would be great if someone could help me out. Thanks!