- #1

joypav

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With this definition the surface area $\omega_{n-1}$ of the sphere in $R^n$ satisfies $\omega_{n-1} = n \gamma_n = \frac{2 \pi^{n/2}}{\Gamma (n/2)}$, where $\gamma_n$ is the volume of the unit ball in $R^n$. Prove that for all non-negative Borel functions $f$ on $R^n$,

$\int_{R^n} f(x) dx = \int_0^{\infty} r^{n-1} (\int_{S^{n-1}} f(r \theta ) d \sigma (\theta) ) dr $.

In particular, if $f$ is a radial function, then ($f$ radial $\implies f(x) = f(\left| x \right|)$)

$\int_{R^n} f(x) dx = \frac{2 \pi^{n/2}}{\Gamma (n/2)} \int_0^{\infty} r^{n-1} f(r) dr = n \gamma_n \int_0^{\infty} r^{n-1} f(r) dr$.

**I admit my head is spinning over this problem. Can someone give me an idea of the proof or explain the idea of it more clearly...**

Also, this problem is in the section for Fubini/Tonelli Theorems.

Also, this problem is in the section for Fubini/Tonelli Theorems.