Consider $$ A:=\left\{(x,y,z)\in\mathbb{R}^3: x^2+y^2\leq 1, 0\leq z\leq 1\right\},\\ v\colon\mathbb{R}^3\to\mathbb{R}^3, (x,yz)\longmapsto (x^3,x^2y,zx^2). $$ Calculate $$ \int_A\langle v,n\rangle\, dS $$ where $n$ is the external unit normal field. Use two ways to do so: 1) surface integral 2) Gauß integral theorem.
1)
I used $$ \Psi\colon [0,1]\times [-\pi,\pi)\to\mathbb{R}^3, (\phi,z)\longmapsto \begin{pmatrix}\cos\phi\\\sin\phi\\z\end{pmatrix} $$ as parametrization and so got $$ \int\limits_0^1\int\limits_{-\pi}^{\pi}\cos^2\phi\, d\phi\, dz=\pi $$
So using surface integral i came to the solution $\pi$. Is that right?
2) How can I use the Gaussian integral theorem to calculate the integral?
I know that $$ \int_A\mbox{div }v\, d^3x=\int_{\partial A}\langle v,n\rangle\, dS $$ but do not know how to work with that here, because it's not the integral over $\partial A$ what is searched but over whole $A$.