I=\int_0^\infty dz\int_0^{2\pi}d\theta\int_{z}^{z\sqrt 2}e^{-r^2}r\,dr=\int_0^\infty dz\int_0^{2\pi}d\theta\int_{z^2}^{2z^2}\frac{e^{-t}}2\,dt=\pi\int_0^\infty(e^{-z^2}-e^{-2z^2})\,dz=\frac{\sqrt\pi^3}{2(2+\sqrt 2)}