Answer
$0$
Work Step by Step
Conversion of rectangular to cylindrical coordinate system is as follows: $r^2=x^2+y^2 \\ \tan \theta=\dfrac{y}{x} \\z=z$
Here, $x=r \cos \theta; y=r \sin \theta, z=z$
As per the problem, $x=\sqrt{4-y^2} \implies x^2=4-y^2$
or, $x^2+y^2=4 \implies r=2$
In the cylindrical coordinate system, we have $I=\int_0^{2\pi} \int_{0}^{2}\int_{r}^{2} (z)(r^2) \cos \theta dz dr d\theta$
or, $I=\int_0^{2\pi} \cos \theta d\theta \times [ \int_{0}^{2}\int_{r}^{2} (z)(r^2) dz dr ] $
or, $I=[\sin \theta]_0^{2 \pi} [ \int_{0}^{2}\int_{r}^{2} (z)(r^2) dz dr ] $
or, $I=[\sin 2 \pi-\sin 0] \times [ \int_{0}^{2}\int_{r}^{2} (z)(r^2) dz dr ] $
or, $I=0$
