Question

Calculus 3 Problem: Use an nᵗʰ-tuple integral to find the volume enclosed by a hypersphere of radius r in n−dimensional space. Hint: The formulas will be different for n even and n odd.

Answer 1

Leading up to this, I had to do this for a circle, then a sphere, and then 4 dimensions. Trying to recognize the pattern...\[\large x_1^2+x_2^2+...+x_n^2=r^2\]
\[\large x_1=\sqrt{r^2-x_n^2-x_{n-1}^2-...-x_2^2}\]