Is there a way to find the volume of the "feasible region" of a standard simplex satisfying simple range constraints?

$x_1+x_2+...+x_n = 1$

$a_1 \le x_1 \le b_1$

$a_2 \le x_2 \le b_2$

$...$

$a_n \le x_n \le b_n$

where $a_i, b_i \in ]0, 1[$, $a_i \le b_i$ are the ranges for each $x_i$.

Of course some of the inequality constraints may be redundant, and the volume can be zero (if there is no solution for the given constraints). Thanks a lot!