Bounding improper integrals from above

Suppose $S$ is a subset of $\mathbb{R}^n$ defined by a system of finitely many polynomial inequalities with integer coefficients. Assume that $S$ is not bounded, but $S$ has finite volume $v$ as a subset of $\mathbb{R}^n$. Can anyone describe an algorithm for generating a sequence of rational numbers that converges to $v$ from above?

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I take it that you are given the inequalities, but you aren't given $v$? –  Gerry Myerson Sep 3 '12 at 2:31
@Gerry Myerson: Exactly. I don't see how to systematically approximate $v$ from above given an arbitrary set of inequalities for which $v$ is finite. –  falang Sep 10 '12 at 22:47