The circular cylinder $\Omega \subset \mathbb{R}^{3}$ with radius $R$ and height $2H$ and the edge $\partial \Omega$ is considered. The cylinder $\Omega$ in a cylinder coordinate system has this form $(r,\phi ,z)\in [0,R)×[0,2)×(-H,H)$.
How do I calculate following integral:
$$\int_{\partial\Omega }^{}2\lambda C\left ( \frac{r}{R^{2}} \left ( 1-\frac{z^{2}}{H^{2}} \right ) + \frac{1}{H^{2}}\left ( 1-\frac{r^{2}}{R^{2}} \right ) \right ) \vec{n}\left ( r,\varphi ,z \right )dA$$