Can anybody give me a hint on this? Let $v\in \mathbb{R}^n,\quad r,R,C$ positive constants . We would like to calculate the volume of the set $$A=\{w\in \mathbb{R}^n : r\leq |w| \leq R, |w-v/2|\leq C \}$$ It seems like the intersection of an annulus and a ball (both in the n-dimensional sense) but i can't write down an integral properly.
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You need to figure out the volume of intersection between two hypersphere and substract.... $$\mu(A) = \mu\left((\bar{B}_R(0) \setminus B_r(0))\cap \bar{B}_C(\frac{v}{2})\right) = \mu\left(\bar{B}_R(0) \cap \bar{B}_C(\frac{v}{2})\right) -\mu\left(B_r(0) \cap \bar{B}_C(\frac{v}{2})\right) $$ – achille hui Dec 27 '18 at 19:46
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To compute the volume of intersection between two hypershere, see answer of this and the wiki entry on hyperspherical cap. – achille hui Dec 27 '18 at 19:54