We can use a nonstandard set of cylindrical coordinates to set up an easy integral. Take
[tex]\begin{cases}x=\xi\\y=r\cos\theta\\z=r\sin\theta\end{cases}[/tex]
and we get
[tex]\displaystyle\iiint_Ez\,\mathrm dV=\int_{\theta=0}^{\theta=\pi/2}\int_{r=0}^{r=12}\int_{\xi=0}^{\xi=\frac13r\cos\theta}(r\sin\theta)r\,\mathrm d\xi\,\mathrm dr\,\mathrm d\theta[/tex]
[tex]\displaystyle=\frac13\int_{\theta=0}^{\theta=\pi/2}\int_{r=0}^{r=12}r^3\sin\theta\cos\theta\,\mathrm dr\,\mathrm d\theta[/tex]
[tex]\displaystyle=\frac16\left(\int_{\theta=0}^{\theta=\pi/2}\sin2\theta\,\mathrm d\theta\right)\left(\int_{r=0}^{r=12}r^3\,\mathrm dr\right)[/tex]
[tex]=864[/tex]
