Past exam of the mathematical methods II course of the University of Oxford / 2019 / 10 / b

On the unit upper hemisphere, use spherical coordinates with $0\le \theta \le \pi /2$ and $0\le \varphi <2\pi$. The surface mass density is $\sigma =z=\cos \theta$, while the surface area element is $dS=\sin \theta d\theta d\varphi$. Thus the total mass isThe answer is in the mass units implicit in the specified unit radius and density law. A useful geometric check is that $zdS$ on the unit hemisphere equals the area element of its vertical projection onto the horizontal unit disc, whose area is $\pi$.