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Consider $0 < r < R$. Our ambient is $\mathbb R^n$ . I want to calculate the integral:

$$ \displaystyle\int_{B(0,R) - B(0,r)}\frac{1}{|x|^n} \ dx.$$

My solution is :

Using the polar coordinates (specifically the coarea formula) we have:

$$\displaystyle\int_{B(0,R) - B(0,r)}\frac{1}{|x|^n} \ dx =\sigma(S^{n-1}) \displaystyle\int_{r}^{R} t^{-n} t^{n-1} \ dt =\sigma(S^{n-1})\ln(R/r).$$

thanks in advance

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    $\begingroup$ I think it's o.k. $\endgroup$ – Christian Blatter Jul 17 '13 at 9:47
  • $\begingroup$ @ChristianBlatter thanks ! $\endgroup$ – math student Jul 17 '13 at 12:55

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