Just a suggestion: adaptive integration. Recursively subdivide the object
into octants, comparing the volume calculated from each subdivision with
the previous stage, and stopping subdivision when the difference in
calculated volumes drops below a certain level.

I've used this technique succesfully in 2D, I don't see any reason for it
not to work in 3D. Adaptive integration is most efficient when each stage
can reuse the results from the previous stage.

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