Suppose $(x_1,\ldots,x_{n+1})$ are coordinates of points on n-sphere. What's the average value of $|x_1|$?
Let $f(n)$ be such value for $n$-sphere. Mathematica suggests that
$$f(10)=\frac{63}{256}$$
What is the formula for general $n$?
Motivation: this predicts behavior of "Kaczmarz" method (also known as ART), which in one case is equivalent to repeatedly projecting a vector onto random hyperplane passing through origin. Value of $f$ indicates how far $w$ moves on average during each projection.
We will use the hyperspherical coordinates outlined here. Let $H^n$ be the $n$-hemisphere, obtained by only letting $\phi_{n-1}$ range from $0$ to $\pi$. In particular, the hypersurface area element $dA$ over $S^n$ or $H^n$ is given by $$dA=\prod_{i=1}^{n}\sin^{n-i}\phi_id\phi_i$$ while $x_n$ is given by $$x_n=\prod_{i=1}^{n}\sin\phi_i.$$ Then, $$f(n)=\frac{1}{\text{Area}(S^n)}\int_{S^n}|x_n|dA=\frac{2}{\text{Area}(S^n)}\int_{H^n}x_ndA=\frac{2}{\text{Area}(S^n)}\int_{H^n} \prod_{i=1}^{n}\sin^{n+1-i}\phi_id\phi_i$$
However, $$\text{Area}(S^{n+1})=2\int_{H^{n+1}} \prod_{i=1}^{n+1}\sin^{n+1-i}\phi_id\phi_i=2\int_0^{\pi}\left(\int_{H^n} \prod_{i=1}^{n}\sin^{n+1-i}\phi_id\phi_i\right)d\phi_{n+1}=\pi \text{Area}(S^{n})f(n)$$
Hence, $$f(n)=\frac{\text{Area}(S^{n+1})}{\pi\text{Area}(S^{n})}=\frac{\Gamma(\frac{n+1}{2})}{\pi^{1/2}\Gamma(\frac{n+2}{2})}$$