Area using Parametric Equations

In summary, to find the area inside the loop of the given curve, we can use the formula for integral and plug in the given functions. However, we need to find the values of alpha and beta, which can be done by solving simultaneous equations for x and y. One approach is to plot points and make a guess, but we can also observe that repeating values of x require equal and opposite t values, while repeating values of y require y=0. Using this information, we can find the values of alpha and beta.
  • #1

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!
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  • #2
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:


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?

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