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I want to calculate volume of the intersection of $n$-simplex $\left\{ x_1,x_2, \ldots, x_n: \sum\limits_{i=1}^nx_i\leqslant 1, x_i\geqslant 0 \right\}$ and $n$-sphere with center in $\left( 1,0,0,\ldots, 0 \right)$ and radius $R \leqslant \frac{\sqrt{2}}{2}$, but really has no idea how.

Thanks for any advice!

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