I have seen some people try to blindly use the formulae for volumes of revolution by cylindrical cross-sections and by cylindrical shells, and I thought that I would write a guide as to how I would recommend tackling such problems, as generally just using the formulae will lead you down blind alleys.
I’ve created an example, with an animation, which I hope will help to master this technique.
So, here is a relatively fool-proof strategy:
- Draw the region which you are going to have to rotate around some axis. This will generally be a matter of:
- Drawing the curves that you have been given
- Finding where they intersect
- Draw the line about which you are supposed to rotate the region
- Draw the reflection of the region about the line of rotation: This gives you a slice through the volume that will be formed
- Now you have to decide which method to use:
- Take a slice through the volume perpendicular to the axis of rotation.