Root Pi from a Bell Curve, by Leaving the Number Line
There is no elementary antiderivative to evaluate, and the checkable slice of that is one line: if p is a polynomial then p' - 2xp has degree deg p + 1, which can never equal the degree of 1. Squaring the integral turns it into a rotationally symmetric integral over the plane, where the polar area element supplies the factor r that makes the radial integral elementary, so I squared equals 2 pi times one half. The same idea survives without polar coordinates via the substitution y = xt, and it fails for e to the minus x to the fourth because x^4 + y^4 is not a function of the radius.