After searching for several hours in the net and on stackexchange and finding absolutely nothing useful for my task:
I have a $A\subseteq \mathbb R^n $ compact with smooth boundary and $x=(0,...0)^T\in A$. Now $\mathcal v(x)$ is the normal unit surface of $A$ and $\alpha(x)=\sphericalangle (x,\mathcal v(x))$ of the position vector $x$ and the normal unit surface $v(x)$. I need to show that for $\mathfrak w_d$, the volume of $\mathbb S^{n-1}\subseteq \mathbb R^n$ follows: $$\int_{\partial A}{\frac{cos(\alpha (x))}{\|x\|^{n-1}}}dS(x)=\mathfrak w_d $$ The hint is stated by observing: $$A_\epsilon =\{x\in A|\|x\|>\epsilon\},\epsilon >0 $$ Is $A_\epsilon =\{x\in A:\|x-0\|>\epsilon\}$ the outside of $\mathbb S^{n-1}=\{x\in \mathbb R^n:\|x\|=1\}$? I hope you can help me, because I don't even have a clue.
Thanks for reading.