Motivation for volume estimation What are some concrete and compelling applications for estimating the volume of convex polyhedra of the sort considered in the more recent papers on random walk methods? 

These papers on volume estimation mention numerical integration as one motivation. What are examples of integrals that people want to compute in practice which are very hard to compute using previous methods?  Or is there some other compelling practical application for computing the volume of a 1000-dimensional polytope?
Estimating the volume of a convex polytope and the closely related task of sampling from it have applications in private data release. 

Roughly, the problem you want to solve is: given a collection of numeric valued queries on a database, come up with answers to those questions that are as close as possible to the real answers, while satisfying differential privacy. In some range of parameters, the optimal algorithm for solving this problem has a geometric description, and implementing it involves sampling from a convex polytope. See here: http://arxiv.org/pdf/0907.3754v3.pdf