 Solution

Let $J={\int }_0^{\infty }{e}^{-x^2}dx$. It is positive and finite: on $(0,1)$ the integrand is at most one, and on $[1,\infty )$ it is at most $e^{-x}$. Since the planar integrand is nonnegative and measurable, Tonelli's theorem givesThe change of variables formula in polar coordinates, with $x=r\cos \theta$, $y=r\sin \theta$ and absolute Jacobian determinant $r$, yieldsOne can apply the formula first on $\delta <r<R$, $0<\theta <\pi /2$, where the coordinate map is a diffeomorphism, and then exhaust the quadrant using the monotone convergence theorem. Taking the positive square root proves the half-line Gaussian integral:Here $dx$ is understood in the one-dimensional integral; it is missing from the printed question.