Evaluate $ \int_{\mathbb{R}^n} \! \exp(-||x||^2) d\mu$

Question

I have to evaluate $ \int_{\mathbb{R}^n} \! \exp(-||x||^2) d\mu$ as part of another problem.

Can someone give me a hint on how to do this?

Answer 1

$$\int_{\mathbb{R}^n}e^{-\|x\|^2}\,dx =\int_{0}^{+\infty}e^{-r^2}\cdot\mu\{x\in\mathbb{R}^n:\|x\|=r\}\,dr=C_n\int_{0}^{+\infty}r^{n-1}e^{-r^2}\,dr$$ where $$ C_n = \frac{n \pi^{n/2}}{\Gamma\left(\frac{n}{2}+1\right)} $$ is the surface area of the unit ball. Since: $$ \int_{0}^{+\infty}r^{n-1}e^{-r^2}\,dr =\frac{1}{2}\,\Gamma\left(\frac{n}{2}\right)$$ we simply have:

$$ \int_{\mathbb{R}^n}e^{-\|x\|^2}\,dx = \pi^{n/2}.$$

That also follows from Fubini's theorem, since: $$ \int_{-\infty}^{+\infty}e^{-x_i^2}\,dx_i = \sqrt{\pi}.$$