While working through the MIT ocw lectures on multi-variable calculus (Prof. Denis Auroux, here), I particularly enjoyed his discussion about the cycloid. Above is a graphic from the wikipedia article (actually the graphic is an animated gif, but I grabbed one of the frames). The red curve is generated by the motion of a point on the edge of a rolling circle.
In addition to the beauty of the curve, it turns out that the length and area under the curve have simple values that are relatively easy to calculate. See wikipedia for the details.
One thing the article doesn't explain is how to get the "parametrization" for the curve. This looks hard, but is made easy by using vectors. It's explained in the second half of Auroux's fifth lecture.
Another thing the article doesn't explain is how to integrate
√(2 - 2 cos t)
Start from the double angle formula:
cos 2s = cos2s - sin2s cos 2s = 1 - 2 sin2s 2 sin2s = (1 - cos 2s) 2 sin2(t/2) = (1 - cos t)
It's straightforward from there.
The Mathworld article is also quite nice, and references a famous challenge in history, the one which led to this quote (in reference to Newton):
"Ah, I know the lion by his paw!"