Suppose that $X \in \mathbb{R}^n$ is a multivariate normal with a covariance given by the identity $I$ . Also, let $S_r$ be an $n$- dimensional sphere with radius $r$. How to compute $Pr[ X\in S_r]$?
In my attempt \begin{align} Pr[ X\in S_r]=\int_{S_r} \frac{1}{\sqrt{(2 \pi)^n n}} e^{-\frac{1}{2} x^T x} dx = \int_{S_r} \prod_{i=1}^n \frac{1}{\sqrt{2 \pi}} e^{-\frac{x_i^2}{2} } dx_1dx_2...dx_n \end{align}
However, I am not sure how to do the above integration. Specifically, how to write limits of each integral. I think the answer should be roughly $1- epx(-r^2/n)$
\begin{align} \textsf P(X\in S_r)=\frac{\int_0^r\exp\left(-\frac12x^2\right)x^{n-1}\mathrm dx}{\int_0^\infty\exp\left(-\frac12x^2\right)x^{n-1}\mathrm dx}=\frac2{2^{\frac n2}\Gamma\left(\frac n2\right)}\int_0^r\exp\left(-\frac12x^2\right)x^{n-1}\mathrm dx\;. \end{align}
$x$ is the radial coordinate, I just couldn't call it $r$ because you'd already used that. You can perform the integration by repeatedly integrating by parts.