I need help with the calculation of the following integral
$ \int_{\mathcal{S}}(1-x_1^2)^rdx_1\ldots dx_n $ where $r>0$ and $ \mathcal{S} = \left\{(x_1,\ldots,x_n):a-\epsilon\leq x_1+\ldots+x_n\leq a,\;|x_i|\leq1 \;\;\forall i=1,\ldots,n\right\} $ for $\epsilon>0$.
This question apparently related to the answer of Fabian in Calculating $\int_{\mathcal{S}}x_1^r \, \mathrm dx_1\ldots \, \mathrm dx_n$, however, in this case there are some non analytically integrals in the way...
Thank you