The Shortest Distance Between 2 Points On The Earth 
Assuming that the earth is a perfect sphere with radius 6378 kilometers, what is the expected straight line distance through the earth (in km) between 2 points that are chosen uniformly on the surface of the earth?

You may assume that the first point is the north pole $N$ ($\theta=0)$ of the earth sphere $S$. All points in an "infinitesimal lampshade" at latitude $\theta$ and of latitude width ${\rm d}\theta$ have the same distance from $N$, namely $$\rho(\theta):=2r\sin{\theta\over2}\ .$$
The area of this "infinitesimal lampshade" amounts to $${\rm d}A=2\pi r\sin\theta\cdot r{\rm d}\theta\ .$$ In this way the mean distance $\bar\rho$ becomes
$$\bar\rho={1\over{\rm area}(S)}\int_S\rho(\theta)\>{\rm d}A={1\over 4\pi\>r^2}\int_0^\pi 2r\sin{\theta\over2}\ 2\pi r^2 \sin\theta\>d\theta\ .$$
As
$$\int_0^\pi \sin{\theta\over2}\sin\theta\ d\theta=2\int_0^\pi \sin^2{\theta\over2}\cos{\theta\over2}\>d\theta={4\over3}\sin^3{\theta\over2}\Biggr|_0^\pi={4\over3}$$
we finally obtain
$$\bar\rho={4\over3}\>r\ .$$
