Derive the formula for the 4-D volume of a hypersphere.
(In reply to Generalized answer: Derivation of volume of a n-dimensional hypershere
The trick is computing V(2n) and V(2n+1), through a lot of messy computation, which I will reluctantly perform it if anyone is curious. one arrives at the eventual result that where V(n) = k(n)r^n, when n is even, k(n) = (π)^(n/2)/(n/2)! and when n is odd, k(n) = 2^n*(((n-1)/2)!/n!)*π^((n-1)/2) when n is even. Start at n=2 a circle, k(2) = π and V(2) = π*r², at n=3 a sphere, k(3) = 4/3π and V(3) = 4/3πr³, at n=4 a "4d-sphere", k(4) = (π²/2) and V(4) = (π²/2)*r^4. For those interested k(5) = (8/15)π² and V(5) = (8/15)π²*r^5.