Evaluate the flux of $F=<x^3,5,z^3>$ through the boundary of the sphere $x^2+y^2+z^2=81$.
It shouldn't be hard to evaluate the volume integral of the sphere as you can set up spherical coordinates. However, the divF is $3x^2+3z^2$ and thus I don't feel like that makes this integral actually easy. I feel like there must be some trick to make this flux easier to compute. Does anyone see it?
Your idea of integrating the divergence over the sphere using spherical coordinates seems right to me. If it seemed complicated to you, perhaps you didn't choose the axes suitably. If you make $y$ the principal axis, with $x^2+z^2=r^2\sin^2\theta$, you get
$$ 3\int_0^Rr^2\mathrm dr\int_0^\pi\sin\theta\,\mathrm d\theta\int_0^{2\pi}\mathrm d\phi\,r^2\sin^2\theta=3\int_0^Rr^4\mathrm dr\int_0^\pi\sin^3\theta\,\mathrm d\theta\int_0^{2\pi}\mathrm d\phi=3\cdot\frac{R^5}5\cdot\frac43\cdot2\pi=\frac85\pi R^5\;. $$