I am interested in computing the following integral which is somewhat similar to computing volume of simplex in $n-$ dimensions :

Let $S_a=\bigg\{(x_1,x_2,\ldots,x_n):\ 0 \leq x_i \leq 1,\ n- \frac{1}{a} \leq \sum_{i=1}^n x_i \leq \frac{1}{a} \bigg\}$ for some $a>0$. I want to evaluate $$ \iint_{S_a}1. \ dx_1 dx_2\ldots dx_n $$

My attempt: The set $S_a$ defined above looks somewhat similar to simplex except the lower and upper limits for the summation. I am trying to find some linear transformation to get rid of them but unable to find so. Can anyone please help me with this ?

  • $\begingroup$ Do you mean $n- \frac{1}{a} \ge \sum_{i=1}^n x_i \ge \frac{1}{a}$? $\endgroup$ – Greg Martin Sep 25 '15 at 22:19
  • $\begingroup$ @GregMartin: No it's the same as in question. We can consider cases to make analysis easy like $1/n \leq a \leq 2/n $ $\endgroup$ – pikachuchameleon Sep 25 '15 at 22:39

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