The open ball of radius $r$ in $\mathbb R^N$ is the set $\{(x_1,x_2,\ldots,x_n)\in R^N \mid \sum_{i=1}^N x_i^2< r^2\}$
By definition its volume $V_N(r)={\int\int\cdots\int} 1 \, dx_1 \, dx_2\cdots dx_N$
How to prove that
$$ V_N = V_{N-1} \int _0 ^{\pi/2} \cos^n \theta \, d\theta$$
V$_n$ is the volume of the n- dimensional unit ball.
Any idea how I can show this please.I have no idea what sort of approach I should take
Best Answer
Search:
"Finding Volume and Surface Area of Hyperspheres in ${\Bbb R}^n$" (Mario Sracic)
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