Given $\phi\in C^1(R)$, and we define the curve and surface $\gamma=\{(x,y):y=\phi(x),0\le x\le 1\}$; $S=\{(x,y,z):z=\phi(\sqrt{x^2+y^2}),x^2+y^2\le 1\}$
I need to prove that $A(S)=2\pi\int_\gamma xdl$, when $A(S)$ is the area of $S$
I need to find the flux transmitted through $S$ down by the vector field
$$F(x,y,z)=\left(\sqrt{x^2+y^2+z^2},\dfrac{1}{\sqrt{x^2+y^2+z^2}},x^2+y^2\right)$$
I don't understand how to this. I tried with Gauss theory but I didn't know from where I get the normal $n$.