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.
- See if this slice is labeled naturally by its position along the x-axis or along the y-axis.
- See whether, for this slice, you will need to take any inverses of functions to calculate the radius of that slice. If so, then you might want to try and use cylindrical shells.
- If you decide that you need to use cylindrical shells, draw in the thin wedges which will be the intersection of the cylindrical shell with the slice that you have after step 3.
- Write down the height of the cylindrical shell in terms of the position labelling the shell.
- Use the formulae for the disk/annulus/thin shell, along with the radii, height and width which you’ve calculated previously to calculate the volume of that one disk/annulus/shell.
- Write down the total volume as the sum of these disks/annuli/shells.
- Take the limit as the width of them goes to zero to give you an integral.
- See what the label in the x or y direction of the first slice/shell is, and the last slice/shell. These will be your integration limits.
- Write out the definite integral and calculate it.
We will look at this with an example (in this case, using a disk/annulus is a good strategy).
Here we will look at two curves, drawn here in red and blue. We will rotate them around the line y=-1. The first thing to do then is to work out what the region looks like:The equations don’t matter for the visualisation, but of course for the integral itself it would be very important. Now we draw in the line of rotation.
From this, we can work out the radius of the inner and outer edges of the annulus as a function of the position of the annulus in the x direction, and we can calculate its area. Calling its width we can calculate its volume. Then we can think of adding up all the annuli. Here we fill in the entire region with annuli: