VERY HARD Puzzle: What Is The Distance Between Two Random Points In A Square?

What is the average distance between two randomly chosen points in a square with a side length 1? The video presents how to estimate the answer and also derives the exact answer using integrals. Per suggestions in the comments, I have modified the title to indicate the difficulty level is VERY HARD (a college level calculus course and probability theory course are necessary background).

Blog post
http://wp.me/p6aMk-4FA

Simulation
https://docs.google.com/spreadsheets/...

Sources and further reading

Quadruple integral
http://math.stackexchange.com/questio...

Difference of two uniform variables
http://www.math.wm.edu/~leemis/chart/...

Difference of two uniform variables and solving the double integral
https://www.quora.com/How-can-I-calcu...

Solving the double integral
http://math.stackexchange.com/questio...

Mean distance polar coordinates change
http://www.mathpages.com/home/kmath32...

Integral secant cubed
https://en.wikipedia.org/wiki/Integra...

Mean distance square
http://mathworld.wolfram.com/SquareLi...

Thanks to Patrons! A video like this takes a lot of research and you guys make it possible.
Kyle
Alberto Nishikawa
Brian M. Mooney

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