Volume of a sphere (r=2a) with hole(r=a) drilled through centre, using spherical polar coordinates.

Question

Need help solving 11.bi),

A cylindrical hole of radius a is bored through the center of a sphere of radius 2a. Find the volume of the remaining material, using spherical polar coordinates. (You should align the bored cylinder with the z-axis).

Have solved the rest of the problems, apart from 11.bi) . I can easily use spherical coordinates to find the volume of the sphere. However I can't work out how to find the volume of the bored out material using spherical coordinates. (Spent quite a few hours trying to work it out).

See diagram/attached picture below for more information.

Any hints/ways to start the problem, define the regions for the bored out volume would be much appreciated.

Thank you for your time, Good Luck!

Answer 1

Look at this plot in $1/8$ of all space, $x,y,z\ge0$:

We have to find two volumes in this part of $\mathbb R^3$. Once, you get the whole volume of the following volume inside the sphere, you can multiply it by $8$ and then subtract it from $4/3\pi(2a)^3$. But about this below shape:

$$V_1: \phi|_0^{\pi/6}, ~~\theta|_0^{\pi/2},~~\rho|_0^{2a}\\ V_2: \phi|_{\pi/6}^{\pi/2}, ~~\theta|_0^{\pi/2},~~\rho|_0^{a\csc(\phi)}$$

Answer 2

The hole is all points $r \sin \theta \lt a$, so that should set the limits of your integration.