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.
(In reply to
re: A solution; Not quite by Larry)
A solution  quite
Rotating clockwise 45 deg maps x > x/Sqrt[2]  y/Sqrt[2], and
y > x/Sqrt[2] + y/Sqrt[2].
Substituting in the original equation gives Sqrt[2]*x^3 + 3*y^2 + 3*Sqrt[2]*x*y^2 = 3*x^2 which is symmetrical about the xaxis.
For y = 0, x =0 and 3/Sqrt[2].
Integrating Sqrt[(3*x^2  Sqrt[2]*x^3)/(3 + 3*Sqrt[2]*x)] between these values gives half the area (3/4).
So the whole area is 3/2?

Posted by goFish
on 20051216 11:59:29 