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Let $||x||$ be any norm over $\mathbb R^n$. Let $B_T$ the open ball with radius $T$ w.r.t. to our norm, i.e. all $x\in\mathbb R^n$ such that $||x||<T$. Let $n\in\mathbb N$.

How much is:$$\intop_{B_T} \sin^2 \frac {||x||\pi(n+\frac 1 2)} T dx$$ and $$\intop_{B_T} \sin^2 \frac {||x||\pi n} T dx$$

At least, how much is it with $L_2$ norm $||x||=||x||_2$?

EDIT: Note that we're actually interested in $L_2$ norm of a function. So maybe can one come with a Fouerier-Transform-Isometry argument to calculate it more easily.

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I would say problem will have a cleaner presentation with $T=1$. Is there a reason an open ball is used instead of a closed one? – Maesumi Jan 11 '13 at 15:19
I agree it'd be cleaner, but in fact I do need that $T$. As for the ball, it can be either open or closed. – Troy McClure Jan 11 '13 at 15:25
For $L_2$ norm you can use spherical coordinates and it is doable, even though a bit lengthy. One of the integrals will be $\int r^{n-1}\sin^2 r dr$ – Maesumi Jan 11 '13 at 15:27
interesting approach – Troy McClure Jan 11 '13 at 15:28

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