Let n be a non-zero natural number, and p a real number, both fixed.
Let $M_k$ ={$x\in R^n | 4^k<=||x||<4^{k+1}$} and $I_k=\int\limits_{M_k} ||x||^p |d^nx|$ for any natural k.
Prove that there is a real $\alpha$ (in terms of n and p) such that $I_6=\alpha I_1$.
I don't really know how to go on about this exercise. Any hints on how to get started?
$I_1> 0$ so we can take $\alpha =\frac {I_6} {I_1}$. This proves existence of $\alpha =\alpha (n,p)$. If you want to explicitly compute $\alpha$ use polar coordinates: Exercise 6 in the chapter on On Integration on Product Spaces in Rudin's RCA.