$Lemma:$Forall $t>0$ we have $\int_{|y| \geqslant t} \frac{dy}{|y|^{n+1}}=\frac{n \omega_n}{t}$, where $\omega_n=\frac{\pi^{n/2}}{\Gamma( \frac{n}{2} +1)}$ is the volume of the $n-$dimensional unit euclideian ball.
This lemma is used for proving that an approximation of unity is a family of good kernels.
$Proof:$
We integrate in polar coordinates writing $y=s \theta$ where $\theta =S^{n-1}$ and $s \in [t, +\infty)$, we have $$\int_{|y| \geqslant t} \frac{dy}{|y|^{n+1}}=\int_{S^{n-1}} \int_{[t, +\infty)}\frac{s^{n-1}}{s^{n+1}}dsd \theta=|S^{n-1}| \int_{[t, +\infty)}\frac{1}{s^2}ds=\frac{|S^{n-1}|}{t}$$
where $|S^{n-1}|=n \omega_n$ is the area of the unit euclideian sphere.
Can someone explain why the proof goes like this and also why the volume of the n dimensional euclideian ball is $\omega_n$?
Thank you in advance!
You are using polar coordinates link. There is something wrong in what you wrote. You should have $$\int_{S^{n-1}}\int_t^\infty\frac{s^{n-1}}{s^{n-1}}\,dsd\theta=\int_{S^{n-1}}\int_t^\infty\frac{1}{s^2}\,dsd\theta=\frac1t\int_{S^{n-1}}1d\theta=\frac1t|S^{n-1}|.$$ As for the unit ball $B(0,1)$ again by spherical coordinates you have $$\omega_n=\int_{B(0,1)}1\, dx=\int_{S^{n-1}}\int_0^1 s^{n-1}\,dsd\theta=\int_{S^{n-1}} \left[\frac{s^{n}}{n}\right]_0^1\,d\theta=\frac1n|S^{n-1}|.$$