Area using Parametric Equations

[Mandanesss]

Homework Statement

Notice the curve given by:
f(t) = x = 36-t^2
g(t) = y = (t^3)-25*t
The curve makes a loop which lies along the x-axis. What is the total area insde the loop.

Homework Equations

Integral from alpha to beta of g(t)*f'(t) dt

The Attempt at a Solution

Ok, so I can easily plug in g(t) and f'(t) into the formula to find the integral. The problem is, I don't understand how to find alpha and beta. I know that if I find where t intersects itself, then I could the alpha and beta. But how do I know where t intersects itself? Please help me out!

 

[Dick]

You want to find two different values of t that give the same values for x and y. So if t=a, and t=b are those values you want to solve the simultaneous equations:

36-a^2=36-b^2
a^3-25*a=b^3-25*b

This is very difficult to do systematically. Your best bet is to plot some points and take a guess. In this case you could observe that x(t)=x(-t), repeating values of x need equal and opposite t's. One the other hand y(t)=-y(-t). So the only way you could get a repeating value of y at the same t that x repeats is for y(t)=0. Can you find a pair of t values?