How to compute integration on sphere?

Question

Let $r>0$ and $A= \{x\in \mathbb R^n: |x|=r\} .$

How to compute $I_1=\int_{A} e^{-a |x|^2} d\sigma (x)$ where $\sigma$ is a surface measure on $A$? Or Can we say that $I\leq C e^{-ar^2} r^{n-1}$ where $C$ is some constant.

If we take $I_2=\int_A |x|e^{-a|x|^2}$, then what can we say about the bound of $I_2$

Answer 1

The function you integrate is a constant here, since $|x|^2=r^2$ on $A$. So $I$ is just equal $e^{-ar^2} \omega_n r^{n-1}$ where $\omega_n$ is a constant independent of $r$. In the second case, it's just $r e^{-ar^2} \omega_n r^{n-1}$

Answer 2

For $x\in A =\Bbb S^{n-1}(r)$ we have $|x|= r$ then

$$I_1=\int_{A} e^{-a |x|^2} d\sigma (x) = \int_{\Bbb S^{n-1}(r)} e^{-ar^2}d\sigma (x) = |\Bbb S^{n-1}(r)| r e^{-ar^2} =|\Bbb S^{n-1}|r^{n-1}e^{-ar^2} $$ $$I_2=\int_A |x|e^{-a|x|^2}d\sigma (x) = \int_{\Bbb S^{n-1}(r)} r e^{-ar^2}d\sigma (x) =|\Bbb S^{n-1}(r)| =|\Bbb S^{n-1}|r^{n} e^{-ar^2} $$