The curve defined by the relation x^3+y^3=3xy intersects itself at the origin and forms a loop. Find the area enclosed by the loop.

Putting x=rCos(t), y=rSin(t) and doing the easy algebra gives

r= 3Cos(t)Sin(t)/(Cos(t)^3+Sin(t)^3)

as the polar coordinate equation for the curve. Using the tools of

http://www.math.tamu.edu/~tom.kiffe/Tools/tools.html

to plot the curve and numerically integrate r^2/2 from t=0 to pi/2, the area of the loop is found to be 1.5 plus or minus some small amount.

Posted by Richard on 2005-12-17 02:44:28