Problem of the Week 1120

Squaring the Cycloid

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Solution

The two areas are the same, as revealed by calling on the integral for an area of a parametrically defined curve.

While this solution is not very interesting, the fact that this relationship was just discovered is moderately interesting. More details of a geometric approach and additional more general results about cycloidal areas can be found in "New insights into cycloidal areas," by T. Apostol and M. Mnatsakanian, Amer. Mathematical Monthly 116 (Aug Sept 2009) 598-611.

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© Copyright 2009 Stan Wagon. Reproduced with permission.