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
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 20051217 13:31:23 