Past exam of the mathematical methods II course of the University of Oxford / 2014 / 10 / a / Solution

The divergence theorem states ${\oint }_{\partial V}\mathbf{F}\cdot \hat{\mathbf{n}}dS={\int }_V\nabla \cdot \mathbf{F}dV$ with outward orientation. For $\mathbf{F}=(x,y,0)$, its divergence is $1+1=2$. The enclosed region has cylindrical bounds $0\le r\le 2$, $0\le z\le 4-r^2$, $0\le \varphi \le 2\pi$, so its volume isThereforeThe planar base contributes zero because its normal is vertical and $F_z=0$; the curved part accounts for the entire outward flux.