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

f(x) ≥ 0 on the interval 0 ≤ x ≤ 1;

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

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.

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.]