Surface Integrals, Stokes & Divergence · Challenge problem

Let $S$ be the surface of the cube $0 \le x, y, z \le 1$ with the bottom face ($z = 0$) removed: the five remaining faces, oriented with outward normals. For $\mathbf F = (xy^2,\ yz + x^2,\ xz^2)$, find $\iint_S (\nabla\times\mathbf F)\cdot d\mathbf S$.