Question

Use a triple integral to find the volume of the given solid. the solid enclosed by the cylinder x2 + y2 = 9 and the planes y + z = 14 and z = 2.

Answer 1

Using cylindrical coordinates, we have

$\begin{cases}x=r\cos\theta\\y=r\sin\theta\\z=\zeta\end{cases}$

so that the Jacobian gives

$\mathrm dV=\mathrm dx\,\mathrm dy\,\mathrm dz=r\,\mathrm dr\,\mathrm d\theta\,\mathrm d\zeta$

The volume is given by the triple integral

$\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=3}\int_{\zeta=2}^{\zeta=14-r\sin\theta}r\,\mathrm d\zeta\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=3}r(14-r\sin\theta-2)\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=3}(12r-r^2\sin\theta)\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle\int_{\theta=0}^{\theta=2\pi}(54-9\sin\theta)\,\mathrm d\theta$
$=108\pi$