I^2=\int_0^\infty\int_0^\infty e^{-(x^2+y^2)}\,dx\,dy=\int_0^{\pi/2}\int_0^\infty e^{-\rho^2}\rho\,d\rho\,d\thet=\frac\pi 2\int_0^\infty\frac{e^{-t}}2\,dt=\frac\pi 4