Arc Length Contest Abstract

Larry Riddle
Agnes Scott College

To enliven our discussion of arc length problems, I often challenge the students in my calculus course to an arc length contest. Each student (or team of students) is asked to find three examples of a continuous function f that satisfies
  1. f(x) ≥ 0 on the interval 0 ≤ x ≤ 1;

  2. f(0) = 0 and f(1) = 0;

  3. the area bounded by the graph of f and the x-axis between x = 0 and x = 1 is equal to 1.

The student must compute the arc length for each of her three functions, and the winner of the contest is the one who has the function with the smallest arc length on the unit interval. This is an interesting open-ended problem for most students since they do not have much experience in constructing functions that satisfy required conditions. I am often surprised by some of the ingenuity students use in coming up with their contest entries. The contest requires evaluation of two different integrals for each function. Moreover, it is important for the students to recognize when one of their functions gives rise to an improper integral for the arc length since such an integral may be difficult to approximate accurately. Thus the contest provides an opportunity to combine several topics from the typical calculus syllabus.

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Published in the College Mathematics Journal, Vol 29, No. 4 (September 1998), 314-320. [Note: The article gives an incorrect value for the arc length of the semi-ellipse, which should be 2.91946 rather than 2.91902.]