A hard $n$-fold integral
Your problem is a special case of the following: given a convex polyhedron defined by a set of linear inequalities, compute its volume.  I believe that problem has been studied in the literature.

For instance, for related work on computing the volume of convex polyhedra/polytopes, see the following:


Algorithm for finding the volume of a convex polytope
Computing the Volume of a Convex Polytope Using Interval Arithmetic
Polytope volume computation


There is probably lots more work out there that might be relevant, but maybe this gives you a start.  You might start by reading that work to see if there are any techniques you can adapt; and by doing a literature review with the above as an entry point into the literature.