Convert to cylindrical coordinates first, using
[tex]\begin{cases}x=r\cos\theta\\y=r\sin\theta\\z=\zeta\end{cases}[/tex]
Then the volume is
[tex]\displaystyle\int_{\zeta=5}^{\zeta=7}\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=\zeta}r\,\mathrm dr\,\mathrm d\theta\,\mathrm d\zeta[/tex]
[tex]=\displaystyle\pi\int_{\zeta=5}^{\zeta=7}\zeta^2\,\mathrm d\zeta[/tex]
[tex]=\dfrac{218\pi}3[/tex]
