Study of an "unknown curve"

"A chewing gum is stuck to the wheel of a bicycle. As the bicycle rides, the chewing gum traces a curve in the air..."

This activity (pdf here) 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.

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.

Using are assigned the task of drawing a curve on a large sheet of 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 create a curve. (This is actually the geometric definition of the cycloid: the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line).

Several tasks are then explained. 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, students will also be able to check the accuracy of their Riemann sum for the area. But finding the area under the curve analytically is beyond the curriculum in AP Calculus AB.

                                              (download the pdf here)