Using cylindrical coordinates evaluate $\iiint_E \sqrt{x^2+y^2}\, dv$, where $E$ is the region inside the cylinder $x^2+y^2=9$ and between the planes $z=1$ and $z=5$. $
Let $ x=r \cos \theta, \ y=r \sin \theta$ and $z=z$. Then the integral becomes $\int_{z=1}^{z=5} \int_{r=-3}^{r=3} \int_{\theta=0}^{\theta=2 \pi} r \cdot r \, dr \, d \theta \, dz.$ Is it true setting, any help is appreciated.
The integral in the OP is not quite correct; the lower limit on $r$ is "$0$," not "$-3$" since $r=\sqrt{x^2+y^2}\ge 0$.
Then, we have
$$\begin{align} \iiint_E \sqrt{x^2+y^2}\,dv&=\int_1^5 \int_0^{2\pi}\int_0^3 r\,dr\,R\,d\phi\,dz\\\\ &=8\pi \int_0^3r^2\,dr\\\\ &=72 \pi \end{align}$$