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

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

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

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