A recent NY Times puzzle (http://wordplay.blogs.nytimes.com/2011/05/02/numberplay-sameweight/) has a problem asking if areas C and D have the same weight when balanced as in the following picture
A number of solutions have been submitted, but I decided to provide one.
See PlanetPTC for a full solution and downloadable Mathcad worksheet.
The idea is simple: use a double integral to compute both areas/weights:
and moments of segments C and D:
Solve for total moment to be 0
to find the balancing point:
Use the balancing point to verify that the weights are not the same: