Solving robust linear matrix inequalities (LMIs) has long been recognized as an important problem in robust control. Although the solution to this problem is well-known for the case of affine dependence on the uncertainty, to the best of our knowledge, results for other types of dependence are limited. In this paper we address the the problem of solving robust LMIs for the case of polynomial dependence on the uncertainty. More precisely, results from numerical integration of polynomial functions are used to develop procedures to minimize the volume of the set of uncertain parameters for which the LMI condition is violated.