\int_0^\infty e^{-x^2}\,dx\int_0^\infty e^{-y^2}\,dy=\int_0^\infty\int_0^\infty e^{-(x^2+y^2)}\,dx\,dy=\int_0^\infty\int_{-\pi}^\pi r\,e^{-r^2}\,dt\,dr=2\pi\int_0^\infty r\,e^{-r^2}\,dt=\pi