Let $f(r)$ be a function of $r = \sqrt{\sum_{i=1}^{n}x_{i}^{2}}$ and $n\geq3$. How do I solve the following integral:
$$\int_{\partial B(0,\epsilon)}f(r)dS(r)$$ with $dS(r)$ as the element of a n-dimensional ball with radius $\epsilon$ and $f(r)=\frac{1}{r^{n-2}}$.
I don`t really understand how to write the integral above in simpler form. Any hint to convert the integral into simpler form is appreciated.
Since $f$ depends only on $r$, it is constant on the sphererical surface. The value of the integral is then $f(\epsilon)$ times the surface of the sphere, which is $\Omega_n\,\epsilon^{n-1}$, where $\Omega_n$ is the measure of the $n-1$-dimensional sphere.