Surface Integrals, Stokes & Divergence · Challenge problem

Let $\mathbf F = \left(z e^{y},\ x\cos z,\ x^2 + y^2\right)$ and let $S$ be the part of the paraboloid $z = 1 - x^2 - y^2$ with $z \ge 0$, oriented with upward normal. Find $\iint_S \mathbf F\cdot d\mathbf S$. Give an exact answer in terms of $\pi$.