Surface Integrals, Stokes & Divergence · Challenge problem

Let $S$ be the lateral surface of the cone $z = \sqrt{x^2 + y^2}$, $0 \le z \le 2$ (no top), oriented with the normal pointing away from the $z$-axis (outward and downward). Find the flux of $\mathbf F = (x, y, z^2)$ through $S$. Give an exact answer in terms of $\pi$.