The Shell Game

It's been a while since I've written anything here. The usual excuse: too busy with work. Here is a little idea you may like. 

Every year about this time as AP Calculus teachers get ready to teach applications of the definite integral the question comes up about whether to teach the method of cylindrical shells. There are several things to consider.

First, the method is not tested on either AP calculus exam; so you will not put your student at a disadvantage if you omit the topic. Any volume of rotation question on the exam can be done with the disk/washer method. But then it is technique that all the texts cover and most college teachers expect students to know. So if there is time it is worth including. It is probably best to include it when you do other volume problems. Due to time considerations some teachers save the topic until after the exam, during the final weeks of school.

However, what I'd like to discuss here is the fact that you never actually need the method of cylindrical shells. You can always avoid it. That said, let me add that shells is usually faster and more efficient than the idea that follows. The example shows how to work around the shell method (pun).

This technique will work for a function that is one-to-one on the interval of integration. If your function is not one-to-one, the interval must be broken into sub-intervals on which the function is one-to-one.

Here is an example: Find the region enclosed by the graph of the lines x = 1,  y = 1, the y-axis and the graph of

is revolved around the y-axis to form a solid figure. Find the volume of this figure.

Proceding by the disk/washer method the volume is given by

 

Note the dy and the limits of integration that correspond to the dy.

The usual next step is to write the integrand in terms of y so that the antiderivative can be computed. Cubic equations can be solved, but who remembers how? And even so the next example may involve a sixth degree polynomial, so let's assume you cannot solve for x in terms of y.  At this point textbooks give up and recommend the method of cylindrical shells.

But it's not necessary that the integrand be in terms of y, it is only necessary that the integrand contain only one variable.  So consider the substitution:

Making this substitution we get

 

The resulting integrand is a polynomial that can be easily expanded and antidifferentiated. Note the change of the limits of integration to correspond with the dx.

Do AP students need to know this? No. It is more of a curiosity or a little "extra" for them. But it is a nice idea and also illustrates a slightly different substitution.