# Change of variable in $n$ dimensions: $\int_{A\setminus B} \int_B \frac{1}{|x-y|^{n+1}}dxdy$.

I am looking to estimate $$\int_{A\setminus B} \int_B \frac{1}{|x-y|^{n+1}}dxdy$$ where $$B$$ is a ball and $$A$$ is a bounded set.

I tried to use the change of variable $$r = |x-y|$$ then the integral becomes: $$\int_0^R \frac{Cr^{n-1}}{r^{N+1}}dr$$ where $$Cr^{n-1}$$ is the volume of the $$n$$ dim sphere and $$R$$ be the "diameter of A". I wonder if this change of variable is correct?

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• I think it's "correct" in the sense that it gives you an upper bound for the double integral, but note that the integral in $r$ is infinite. Mar 18 at 16:48