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
A solution - quite
Rotating clockwise 45 deg maps x -> x/Sqrt - y/Sqrt, and
y -> x/Sqrt + y/Sqrt.
Substituting in the original equation gives Sqrt*x^3 + 3*y^2 + 3*Sqrt*x*y^2 = 3*x^2 which is symmetrical about the x-axis.
For y = 0, x =0 and 3/Sqrt.
Integrating Sqrt[(3*x^2 - Sqrt*x^3)/(3 + 3*Sqrt*x)] between these values gives half the area (3/4).
So the whole area is 3/2?
Posted by goFish
on 2005-12-16 11:59:29