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Area using Parametric Equations

  1. Mar 21, 2007 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant equations

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

    3. 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!
  2. jcsd
  3. Mar 22, 2007 #2


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    Science Advisor
    Homework Helper

    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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