 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  A Self Intersecting Curve (Posted on 2005-12-16) 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) Comments: ( Back to comment list | You must be logged in to post comments.) re: Numerical Integration Solution | Comment 8 of 9 | (In reply to Numerical Integration Solution by Richard)

The integral for the area using polar coordinates, that is the integral from t=0 to pi/2 of r^2/2, is the integral of a rational function of Sin(t) and Cos(t).  Such integrals can be evaluated in closed form by using the method shown in the web page from sosmath.com entitled "Rational Expressions of Trigonometric Functions." See

http://www.sosmath.com/calculus/integration/raextrig/raextrig.html

This would be a lot of work for this problem, although there may be tricks that would lighten the burden.  However, this gives a way to prove that the answer is EXACTLY 1.5, as the numerical integration SUGGESTS.

 Posted by Richard on 2005-12-17 13:31:23 Please log in:

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