While studying the plume rise Gaussian Model , I came across following Improper integral which I was unable to solve :
$$\int_{0}^\infty \int_{-\infty}^{\infty} \mathrm{exp}[-((y^2+z^2)/2)]\,dy\,dz$$
The source I am referring says to apply the transformation
$y=r \cos θ$
$z=r \sin θ$
I am unable to do solve this integral. Please help me out.
You are transforming the integral to polar coordinates.
If $y=r\cos\theta$ and $z=r\sin\theta$, then $y^2+z^2=r^2$. Also, your region $z \geq 0$ (and $y$ anything) is the upper-half of the $yz$-plane. This is $r \geq 0$ and $0 \leq \theta \leq \pi$ in polar.
Don't forget the Jacobian ($J=r$). Therefore, your integral transforms to $$\int_0^\pi \int_0^\infty e^{-r^2/2} \,r\,dr\,d\theta = -\pi e^{-r^2/2}\Bigg|_0^\infty = \pi$$
Technically, there are issues with transforming your iterated integral to a double integral and then back to an iterated integral (in the new coordinate system). But everything works out since all integrals converge...and converge absolutely since you have a non-negative function.