I am not sure if this is a duplicate, in case it is, please inform me.
I trie to compute the volume of an $n-$ball. That is the volume of $B(0,1)$ in $\mathbb{R}^n$. My intuition was to use the spherical coordinates, which is pretty hard of we do it in any $n \in \mathbb{N}^*$. Now, one the classical ways of computing the volume of an $n-$ball is to introduce the gaussian integral. That is considering the function $$f(x_1,\dots,x_n) = exp (-\frac{1}{2} \sum_{i=1}^{n}x_i^2) $$ and then computing $$\int_{\mathbb{R}^n}fd\mu$$ But this is pretty odd to me, as despite giving the correct result, I still don't understand why it gives the correct result. How come a function $x \mapsto e^{\frac{-1}{2}x^2} $ give the volume of a unit ball? What am I missing? I am pretty sure this must be due to some lack of understanding of the measure theory or integration theory, so could anyone explain why this integral works?