This is problem 8.4.17. from Marsden Vector Calculus book.
Let $\rho$ be a continuous function which vanishes outside a 3D region $W$. Define
$\phi(\textbf{p})=\displaystyle\iiint_W\frac{\rho(\textbf{q})}{4\pi\lVert\textbf{p}-\textbf{q}\rVert}dV(\textbf{q})$.
How to show that
$\displaystyle\iint_{\partial W}\nabla\phi\cdot\textbf{dS}=-\iiint_W\rho\; dV$
without the knowledege that $\phi$ is the solution to the Poisson equation $\nabla^2\phi=-\rho$ (which will be a consequence of this)?