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Alja [10]
4 years ago
7

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.
Mathematics
1 answer:
Alexxx [7]4 years ago
6 0
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
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