What is the formula for the volume of a solid unit n-sphere in n+1 dimensions?

[Euge]

Here is this week's POTW:

-----
Derive a formula for the volume of the solid unit $n$-sphere in $\Bbb R^{n+1}$.

-----

Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!

 

[Euge]

No one answered this week's problem. You can read my solution below.

Spoiler

Let $D^n$ denote the solid $n$-sphere in $\Bbb R^{n+1}$. By slicing, $$\operatorname{Vol}(D^n) = \int_{-1}^1 \left(\sqrt{1-x_{n+1}^2}\right)^{n-1}\operatorname{Vol}(D^{n-1})\, dx_{n+1} = \int_{-1}^1 (1 - x_{n+1}^2)^{\frac{n-1}{2}}\, dx_{n+1}\operatorname{Vol}(D^{n-1}) = \operatorname{Vol}(D^{n-1}) \cdot 2\int_0^{\frac{\pi}{2}} \cos^n\theta\, d\theta$$ The integral $2\int_0^{\pi/2} \cos^n\theta\, d\theta$ is the Beta function $B((n+1)/2, 1/2)$, thus $$2\int_0^{\frac{\pi}{2}}\cos^n\theta\, d\theta = \frac{\Gamma\left(\frac{n+1}{2}\right) \Gamma\left(\frac{1}{2}\right)}{\Gamma\left(\frac{n+2}{2}\right)} = \frac{\Gamma\left(\frac{n+1}{2}\right)\sqrt{\pi}}{\Gamma\left(\frac{n+2}{2}\right)}$$ Hence $$\operatorname{Vol}(D^n) = \operatorname{Vol}(D^0)\prod_{j = 1}^n \frac{\Gamma\left(\frac{j+1}{2}\right)\sqrt{\pi}}{\Gamma\left(\frac{j+2}{2}\right)} = (\sqrt{\pi})^n \prod_{j = 1}^n \frac{\Gamma\left(\frac{j+1}{2}\right)}{\Gamma\left(\frac{j+2}{2}\right)} = \pi^{n/2} \frac{\Gamma(1)}{\Gamma\left(\frac{n+2}{2}\right)} = \frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2} + 1\right)}$$