Using cylindrical coordinates, we have
[tex]\begin{cases}x=r\cos\theta\\y=r\sin\theta\\z=\zeta\end{cases}[/tex]
so that the Jacobian gives
[tex]\mathrm dV=\mathrm dx\,\mathrm dy\,\mathrm dz=r\,\mathrm dr\,\mathrm d\theta\,\mathrm d\zeta[/tex]
The volume is given by the triple integral
[tex]\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=3}\int_{\zeta=2}^{\zeta=14-r\sin\theta}r\,\mathrm d\zeta\,\mathrm dr\,\mathrm d\theta[/tex]
[tex]=\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=3}r(14-r\sin\theta-2)\,\mathrm dr\,\mathrm d\theta[/tex]
[tex]=\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=3}(12r-r^2\sin\theta)\,\mathrm dr\,\mathrm d\theta[/tex]
[tex]=\displaystyle\int_{\theta=0}^{\theta=2\pi}(54-9\sin\theta)\,\mathrm d\theta[/tex]
[tex]=108\pi[/tex]
