Answer
$\dfrac{2\pi}{5}$
Work Step by Step
Conversion of rectangular to cylindrical coordinate system gives: $r^2=x^2+y^2 \\ \tan \theta=\dfrac{y}{x} \\z=z$
Here, $x=r \cos \theta; y=r \sin \theta, z=z$
Let $I=\iiint_Ex^2dV=\int_0^{2\pi} \int_{0}^{1}\int_{0}^{2r} x^2 r dz dr d\theta$
or, $I=\int_0^{2\pi} \int_{0}^{1}\int_{0}^{2r} (r\cos \theta)^2 r dz dr d\theta$
or, $I=\int_0^{2\pi} \int_{0}^{1}[r^3 \cos^2 \theta(z)]_{0}^{2r} dr d\theta=\int_0^{2\pi} \int_{0}^{1}2r^4 \cos^2 \theta dr d\theta$
or, $I=\int_0^{2\pi}(\dfrac{2}{5})\cos^2 \theta d\theta$
or, $I=\dfrac{2}{10}\int_0^{2\pi} (1+\cos 2 \theta) d\theta=\dfrac{2}{10}\int_0^{2\pi} [\theta+\dfrac{\sin 2 \theta}{2})_0^{2\pi}=\dfrac{2\pi}{5}$
