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