Convert to spherical coordinates.
[tex]\begin{cases}x=\rho\cos\theta\sin\varphi\\y=\rho\sin\theta\sin\varphi\\z=\rho\cos\varphi\end{cases}\implies\mathrm dV=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]
[tex]\displaystyle\iiint_E(x^2+y^2)\,\mathrm dV=\int_{\varphi=0}^{\varphi=\pi}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=3}^{\rho=4}\underbrace{\rho^2\sin^2\varphi}_{x^2+y^2}\underbrace{\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi}_{\mathrm dV}[/tex]
[tex]=\displaystyle2\pi\left(\int_{\varphi=0}^{\varphi=\pi}\sin^3\varphi\,\mathrm d\varphi\right)\left(\int_{\rho=3}^{\rho=4}\rho^4\,\mathrm d\rho\right)=\frac{6248\pi}{15}[/tex]
