Shells Examples

Let R be the region surrounded by , the line x = 1, and the x-axis. Use the shell method to find the volume of the solid obtained by rotating R around the line x = 1.

This is the second example we've had using this region, so we know what it looks like by now. The shell at position x will look like

The height of the shell is still . That hasn't changed since the previous example. But the radius of the shell has changed. The radius of the shell is the distance from the edge of the shell to the axis of rotation, and we've changed the axis of rotation. From the picture, we can see that the radius of the shell is (1 – x). The volume of this shell is

We still have shells from x = 0 to x = 1, so these are still the limits of integration.

The volume of the solid is

Even though we're playing with toilet paper tubes instead of frisbees, we're still going to be bombarded with enough examples and exercises to make sure we really understand the method. If we get bored, we can always glue some googly eyes and pipe cleaner to a toilet paper tube and make some TP Peeps.