Setting
[tex]\begin{cases}x(r,\theta,\zeta)=r\cos\theta\\y(r,\theta,\zeta)=r\sin\theta\\z(r,\theta,\zeta)=\zeta\end{cases}[/tex]
you arrive at the Jacobian
[tex]J=\dfrac{\partial(x,y,z)}{\partial(r,\theta,\zeta)}=\begin{vmatrix}x_r&x_\theta&x_\zeta\\y_r&y_\theta&y_\zeta\\z_r&z_\theta&z_\zeta\end{vmatrix}[/tex]
[tex]J=\begin{vmatrix}\cos t&-r\sin t&0\\\sin t&r\cos t&0\\0&0&1\end{vmatrix}=r\end{vmatrix}[/tex]
Then the integral is
[tex]\displaystyle\iiint_E(x^2+y^2)\,\mathrm dV=\int_{\zeta=2}^{\zeta=3}\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=3}r^2|J|\,\mathrm dr\,\mathrm d\theta\,\mathrm d\zeta[/tex]
[tex]=\displaystyle\int_2^3\mathrm d\zeta\int_0^{2\pi}\mathrm d\theta\int_0^3r^3\,\mathrm dr[/tex]
[tex]=(3-2)\times(2\pi-0)\times\dfrac14r^4\bigg|_{r=0}^{r=3}[/tex]
[tex]=\dfrac{81\pi}2[/tex]
