Use the divergence theorem to calculate $\int _TF\cdot n\:dS\:$ when $F(x,y,z)=x^3i+y^3j+z^2k$ and $T$ is the surface area to the half sphere given by $x^2+y^2+z^2\le1$ , $z\ge0$. The normal vector should point out of the half sphere.
I have managed to set up the integral in two different ways,
The first one being: $\int _0^{2\pi}\:\int _0^{\frac{\pi}{2}}\:\int _0^1\:\left(\left(3\left(\rho ^2-\rho ^2\cdot \:cos^2\left(\phi \right)\right)\right)+2\cdot \rho \cdot \:cos\left(\phi \right)\right)\rho ^2sin\left(\phi \right)d\rho \:d\phi \:d\theta $
The second one being: $\int _0^{2\pi}\:\int _0^1\:\int _0^{\sqrt{1-r^2}}\:3r^2+2z\:dz\:r\:dr\:d\theta $
Both of the integrals gives the answer, $\frac{13\pi }{10}$, which is wrong. The correct answer according to my book is, $\frac{6\pi }{5}$.