<Divergence theorem> says $\int_V\nabla\cdot F\,dV=\int_{\partial V}F\cdot n\,dS$ for the outward normal. Here $\nabla\cdot(xe^{-r^2})=(3-2r^2)e^{-r^2}$, so radial integration gives $4\pi a^3e^{-a^2}$. On the sphere $F\cdot n=ae^{-a^2}$, whose <integral> is the same $4\pi a^3e^{-a^2}$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 3 / 4A / Solution

 Solution (source code)