Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 3 / 9A / a / Solution

The divergence theorem is ${\iint }_{\partial V}F\cdot ndS={\iiint }_V\nabla \cdot FdV$. Here $\nabla \cdot F=3z+2yz$. The $y$ term integrates to zero by symmetry, while the ellipse has area $\pi /\sqrt{ab}$. Hence the flux is