Using spherical coordinates to bound $\int_{\mathbb{R}^n} \|x-y\|_2 ^{-a}dy$?

Question

Is it possible to bound $$\int_{\mathbb{R}^n} \|x-y\|_2 ^{-a}dy$$ with $$\int_{\mathbb{R}^n}\frac{r^{n-1}}{r^a}dr$$ by using spherical coordinates?

For $n=3$ this is clear, but what about $n>3$?

Answer 1

Use Hyperspherical_coordinates. See an example of application in Volume of Region in 5D Space.