Let $\Omega={B}_1(0)/B_{\epsilon}(0)\subset\mathbb{R}^n$, $f:\Omega\to\mathbb{R}$ contínuous and $S^1$ be the sphere in $\mathbb{R}^{n-1}$. I want to evaluate the integral $$I:=\int_{\Omega}f(x)dx.$$
My question is:
Is $I$ equal
$$\int_{\epsilon}^{1}\int_{S^1}f(rx)dxdr ?$$
I tried polar coordinates but didn't when too far.