Alphabeta MathHow statement and proof provenance work
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Polar coordinates recover the published ball-volume and Gaussian formulas
Assuming the Axiom of Countable Choice required by Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, applying the polar formula to radial indicators recovers the published volume formulas The closed form for the volume of the unit -ball and The volume of a radius- closed -ball is . The cited The plane Gaussian integral equals by polar coordinates is instead an earlier instance of the same polar method; the present theorem subsumes that two-dimensional calculation rather than providing an independent Gamma-function route to it.
This page records the bridge, not a duplicate proof of those already-published closed forms.
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Sources
- Gerald B. Folland, Real Analysis, 2nd ed., Proposition 2.53 and Corollary 2.55 (standard reference, not scraped)