2.3 Volumes of Revolution: Cylindrical Shells 

P. 166: 120 – 131 all, 140-143 all, 145, 148, 158, 159
P. 271: 61

Webwork: Shells and Washers due 5/17

Notes from lecture and office hours:


Method of cylindrical shells

It will be helpful to remember the pictures from Monday’s lecture.

The method of cylindrical shells is another method for calculating volumes of solids of revolution. It differs from the methods discussed previously in that we’re no longer slicing to get areas of cross sections. (In some sense, though, we still are slicing; but the slices are no longer flat, they’re cylindrical). But the general principal is the same: we’ll integrate 2-dimensional areas to get the 3-dimensional volume.

Remember that vertical segment connecting the x-axis with the graph y=f(x) at position x? When we rotated it around the x-axis, we got a disk. But what shape would we get if we rotate it around the y-axis (a line that’s parallel to the vertical segment instead of perpendicular to it)?

The visualization might take a bit of time if you haven’t tried to see it before. Hold your pencil vertically and then move your hand in a circle that’s parallel to the floor. Imagine you took a stop-motion video of this; what shape would your pencil sweep out?

Well, this method is called “cylindrical shells” because the answer to both of these questions is “a cylindrical shell.” Scroll to Figure 2.26 here. Your cylinder is hollow and has no top or bottom. The area we’re interested in is its surface area. Try cutting this cylinder vertically; what shape do you get?

How can you understand the formula:

V = \int_a^b 2 \pi f(x) dx?

  • This video (8 minutes) takes you through the formula and shows an example where the axis of rotation is the y-axis.
  • This video (9 minutes) does the same thing, but now the axis of rotation is the x-axis.  This approach tends to be helpful if your graph gives x in terms of y: x = f(y).
Which volume method when?

This is the big question. Part of the challenge in calculating volumes is knowing when to use which method. Following these segments through a rotation will help you determine the shape you’re dealing with; it will be a disk, a washer, or a cylindrical shell.