Integrals over the unit square arising in geometric probability are integral_0^1 integral_0^1 sqrt(x^2 + y^2)d x d y = 1/3[sqrt(2) + sinh^(-1)(1)] integral_0^1 integral_0^1 sqrt((x - 1/2)^2 + (y - 1/2)^2)d x d y = 1/6[sqrt(2) + sinh^(-1)(1)], which give the average distances in square point picking from a point picked at random in a unit square to a corner and to the center, respectively.

double integral | Hadjicostas's formula | hypercube line picking | square point picking | unit disk integral | unit square