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Let's recap what you want to do: You have some set $V \subset \mathbb{R}^5$ and want to approximate the integral of some function $f\colon V \to \mathbb{R}$: $$ \int_V f(x) \,\mathrm{d}x $$ The "mean value method" sounds like a Monte Carlo-type approximation of the form $$ \int_V f(x) \,\mathrm{d}x \approx \frac{\operatorname{vol}(V)}{N} \sum_{i=1}^...


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