Suppose $S$ is a subset of $\mathbb{R}^n$ of finite volume defined by a system of finitely many polynomial inequalities with integer coefficients. Can anyone describe an algorithm that, given such a system of inequalities, generates a sequence of rational numbers that converges to the volume of $S$ from above?

This question expands a comment of Andrej Bauer to a related question.

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The case that baffles me is when $S$ is unbounded. The obvious approach would be to find some general way to enlarge $S$ by "a little bit" to a set whose volume is easy to compute. But I don't see how to do this.

Actually, I cannot even describe an algorithm to determine whether or not $S$ has finite volume, and such an algorithm might give a good start to solving the original problem