Algorithm for computing volume of union or intersection of n-dimensional convex polytopes given their facets?
A standard approach is to decompose it into a bunch of triangles (well, the n-dimensional generalization of triangles), then compute the volume of each triangle and sum them up.  Have a look at https://mathoverflow.net/q/979 or https://scicomp.stackexchange.com/q/2881.  Be warned that this could take exponential time.

There are also algorithms for approximating the volume of such a polytope, if that is useful to you.