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Pappus's centroid theorem

Guldin's rule, Guldinus theorem, Guldin's theorem, Pappus's theorem for solids of revolution, Pappus's theorem for surfaces of revolution
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I'm pretty sure that theorem 1 of the present entry
should work also for rectifiable curves, not merely smooth curves[*];
however I don't know about geometric measure theory in order to prove it.
Can anyone confirm if my conjecture is correct?

[*] N.B. Theorem 2 already allows arbitrary Lebesgue-measurable regions;
this is not hard to prove with the classic measure theory tools.

// Steve

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