Surface Integrals, Stokes & Divergence · Challenge problem

Let $S$ be the part of the sphere $x^2 + y^2 + z^2 = 4$ that lies inside the cylinder $x^2 + y^2 = 1$ and above the $xy$-plane, with upward normal, and let $\mathbf F = (-yz,\ xz,\ xy)$. Use Stokes' theorem to find $\iint_S (\nabla\times\mathbf F)\cdot d\mathbf S$. Give an exact answer.