Use Gauss' theorem to the surface integral of an hemisphere

Question

Use Gauss' Theorem to show that $\int_S \,d\vec{\sigma}=0$ if S is a closed surface. Use the result to evaluate $\int_{Shem} \,d\vec{\sigma}$, where $Shem$ is the hemisfere of a sphere of radius $R$.

Answer 1

From the Divergence Theorem, we have

$$\oint_{S} \vec x_i\cdot d\vec\sigma=\int_V \nabla\cdot(\hat x_i)\,dV=0$$.

Also, we have

$$\int_{\text{hemisphere}} d\vec\sigma =\int_0^\pi\int_0^{\pi}(\hat x\sin(\theta)\cos(\phi)+\hat y\sin(\theta)\sin(\phi)+\hat z\cos(\theta))\,\sin(\theta)\,d\theta\,d\phi$$

Can you finish?