How to derive the volume of an n-dimensional hypersphere (the long version)

I was trying to get a better intuition for "the curse of dimensionality" in machine learning, and needed to know the volume of a unit n-sphere -- so I remembered how to derive it. This derivation is quick, easy, and cool, and I want to share it with the world :) This is the detailed version of my 3-minute derivation (   • How to derive the volume of an n-dimension...  )