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A Self Intersecting Curve (Posted on 2005-12-16) Difficulty: 5 of 5
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.

See The Solution Submitted by Brian Smith    
Rating: 4.5000 (2 votes)

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re(2): A solution; Not quite | Comment 3 of 9 |
(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 x-axis.

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 2005-12-16 11:59:29
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