Ranking Volumes of Solids of Revolution

One of the explicit goals of the integral calculus course I am teaching this quarter is for the students to develop geometric visualization skills.  How do you train students to generate mental pictures?  One can:

• give them an easy picture to remember, i.e., do an example where you draw a simple region in the plane like a rectangle in the first quadrant with an edge on the -axis and then draw by hand the cylinder it sweeps out through revolution about the -axis.
• show them a computer animation of more complicated region revolving and generating a solid.
• show them a physical demonstration of the construction, using something like a paper wedding bell.
• show them a familiar rotationally symmetric object being milled on a lathe.

Each of these presentations creates an image in the mind of the student, but ultimately it is the practice of creating similar images on their own through exercises that helps them to develop their geometric imagination.  Unfortunately, students often seek out help with their out-of-class work from friends, tutoring centers, online tutorials, etc., and in doing so may bypass this aspect of the exercise altogether.  To combat that trend, I’ve tried the following in-class activities:

1. On day that solids of revolution are introduced, I do a physical demonstration with a modified paper wedding bell.  I cut the profile to approximate the graph of over , so that when unfurled the “bell” looks like a paraboloid.  After that, I teach them how to draw a picture of the solid, set up an integral to compute its volume, carry out the computation, and then do a sanity check by comparing the computed value to the volumes of an enveloping circular cylinder and an inscribed right circular cone.
2. The next class period, after they have a had some time to absorb the idea and to work with examples, I do a more complicated example with an associated with an animation.  Time permitting, I talk about how we can make such shapes using lathes and show a quick video.  This gives a touch of practical realism to the subject and cements its importance.
3. By the next class period, they are ready to tackle a ConcepTest like this one:

Notice that no equations are given for the curves bounding the region.  This is by design. There are no integrals to set up and calculate in order to answer the question.  What is important is visualization and critical thinking about the comparison.  This kind of question requires a different level of cognitive activity.  Not only do they need to envision the solids in a rough sense, they also need to sharpen their mental pictures with scale in order to facilitate comparison.  This is difficult to do, and students find it challenging or perhaps even too difficult as an individual exercise.  But posing such a question as a ConcepTest, harnesses the power of collaborative learning.  After an initial vote, the small group discussion forces students to verbalize their mental pictures and compare with those of their fellow students.  Through such discussion, their mental images are adjusted, sharpened and uniformized.