Consider the quantity


$$h\_n:=\frac{S\_{n-2}}{V\_{n-2}} \int\_{r=0}^1 r^{n-2} \sqrt{1-r^2} dr$$


where $S\_{n-2}$, $V\_{n-2}$ is respectively the surface and volume of the hypersphere in $\mathbb R^{n-1}$.


**Question:** How can we understand $h\_n$? In particular, is $h\_n$ increasing?
Does it tend to $1$ as $n$ goes to infinity? 


**PS:** The formula i quote above is not the average height of unit hemisphere. By average height of unit hemisphere i mean the integral $\int\_{0}^1 |x\_1| d \mu$, where $\mu$ is the uniform measure on $S^{D-1}$ and $x=(x\_1,\dots,x\_D)$ is the coordinate representation of a point $x \in S^{D-1}$. This integral i have calculated to be $\frac{(D-1)!!}{(D-2)!!} \frac{2}{\pi}$ if $D$ is even, and $\frac{(D-1)!!}{(D-2)!!}$ if $D$ is odd.



understanding the average height of a unit hyper-semisphere
Your expression equals $$\frac{\sqrt{\pi } (n-1) \Gamma \left(\frac{n-1}{2}\right)}{4 \Gamma
 \left(\frac{n}{2}+1\right)},$$ which goes to zero as $n$ goes to infinity.


