Riemann Sums Approximations
Volumes of Revolution
80 minutes in class (no homework)
This activity is short enough to be entirely contained in one block. We don’t ask any write-up of results. Obviously, if we did, more time would be required.
Summary of Activity
“A chewing gum is stuck to the wheel of a bicycle. As the bicycle rides, the chewing gum traces a curve in the air…” Groups of four are assigned the task of drawing a curve on a long sheet of butcher paper that is stuck to the wall, by attaching a felt pen (or whiteboard pen etc.) to the edge of a classroom (circular) trashcan, and roll the trashcan in order to draw the curve. This is actually the geometric definition of the cycloid. Then the students need to approximate the area under one arch the curve using a Riemann Sum. They also need to approximate the volume created if an arch is rotated about the x-axis, again using Riemann Sums. In parallel, students research the internet for information about the curve, in particular its name, and the actual formula for the area under an arch — which they can now compare with their approximation.
Notes and Insights
The “unknown curve” referred to is actually the cycloid. It is assumed that students don’t know this curve at the beginning of the lesson, or at least not very well.
This activity combines hands-on work, research of information on the internet, and applications of the rectangular method of approximation of integrals (Riemann Sums). It is a good and fun activity to apply Riemann sums in a meaningful way, and it will greatly interest students who are curious about math.
This activity is structured as a worksheet with several tasks listed. Students need to estimate the area under the arch using Riemann sums, as well as the volume of revolution that is formed by rotating the curve about the “x-axis”. They will also investigate online to find the name of the curve, historical background and other properties of the curve.
Given that the area under the curve is three times the area of the generating circle (task 3 in the worksheet), students will also be able to check the accuracy of their Riemann sum for the area (task 4). But finding the area under the curve analytically is beyond the curriculum in AP Calculus AB.
This activity can be adapted to extend beyond 80 minutes if students have to write a report. At KIS, we just ask groups to hand in one completed worksheet and their curve.
Given the width of butcher paper, it is possible to cut them in two along the middle and hand out thin long rectangles of paper to each group (about 50cm x 250cm).