Question

Use cylindrical coordinates. find the volume of the solid that lies within both the cylinder x2 y2 = 9 and the sphere x2 y2 z2 = 16.

Answer 1

In Cartesian coordinates, the region is given by $-3\le x\le3$, $-\sqrt{9-x^2}\le y\le\sqrt{9-x^2}$, and $-\sqrt{16-x^2-y^2}\le z\le\sqrt{16-x^2-y^2}$. Converting to cylindrical coordinates, using

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

we get a Jacobian determinant of $r$, and the region is given in cylindrical coordinates by $0\le\theta\le2\pi$, $0\le r\le3$, and $-\sqrt{16-r^2}\le z\le\sqrt{16-r^2}$.

The volume is then

$\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=3}\int_{z=-\sqrt{16-r^2}}^{z=\sqrt{16-r^2}}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta=\dfrac{4(64-7\sqrt7)\pi}3$