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I'm wondering about different ways to compute the volume of an $n$-sphere. Please see the wikipedia page for one method to compute the volume via hyperspherical coordinates:

Suppose now I want to compute the volume $V(n)$ of an $n$-sphere by integrating the volumes $V(n-1)$ of a whole bunch of $(n-1)$-spheres. Assuming that this $(n-1)$-sphere is aligned with the first $n-1$ coordinate axes, I don't see a way to just integrate $(n-1)$-spheres with the variable being the last angular coordinate. I'm also more generally interested in this question when we integrate $(n-k)$-spheres with volumes $V(n-k)$ with the variables being the last $k$ angular coordinates. Any thoughts?

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