Calculus 3 Spherical coordinates: I'm not sure how to set this up.

Question

find the volume of the region enclosed by the sphere $x^2+y^2+z^2=324$ and the cylinder $(x-9)^2+y^2=81$ by using spherical coordinates. I'm just not seeing how to convert this into a form where spherical coordinates are usable. Could someone show me how to set this up? I can handle it from there. Thank you.

Answer 1

It seems the following.

If we use spherical coordinates $$x=r\sin\varphi\cos\theta,$$ $$y=r\sin\varphi\sin\theta,$$ $$z=r\cos\varphi$$ $$(0\le\varphi\le\pi, -\pi\le\theta\le \pi),$$ we obtain the bounds of the integration domain: $$0\le\varphi\le\pi, \pi/2\le\theta\le\pi/2.$$ The lower bound for $r$ is $0$. Let $U\le 18=\sqrt{324}$ be the upper $U$ bound for $r$. Then

$$(U\sin\varphi\cos\theta-9)^2+ (U\sin\varphi\sin\theta)^2\le 81$$

$$U\sin^2\varphi\le 18\sin\varphi\cos\theta$$

$$\sin\varphi=0\mbox{ or }U\sin\varphi\le 18\cos\theta.$$

So we can put $$U=18\min\left\{1,\frac{\cos\theta}{\sin\varphi}\right\}.$$