The problem reads:
Find $ \int_S \, f(x, z) \, dS $, where $ f(x, z) = e^{-(x^2 + z^2)} $ and $S$ is the unit disk centered at the point $(0, 2, 0)$ and in the plane $y=2$.
I'm not sure how to set up this problem because of where the surface is placed in $ \mathbb{R}^3 $.
Parmaterize your surface
$x = r\cos \theta, z = r\sin \theta, y = 2\\ \|dS\| = \|(\frac {\partial x}{\partial r},\frac {\partial y}{\partial r},\frac {\partial z}{\partial r})\times(\frac {\partial x}{\partial \theta},\frac {\partial y}{\partial \theta},\frac {\partial z}{\partial \theta})\| \ dr\ d\theta\\ \|dS\| = r\ dr\ d\theta$
$\iint re^{-r^2} \ dr\ d\theta$