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Homework Help: Area enclosed by the curve

  1. Apr 24, 2007 #1
    The problem is the following:

    Use a line integral to find the plane area enclosed by the curve r = a*(cos t)^3 + b*(sin t)^3, 0 < t <2*pi.

    I don't really have a clue how to solve that. The chapter we are right now in is about Green's theorem in the Plane, and all the example problems use x and y instead of t.

    I would be very glad i f somebody would tell how to solve this problem.
  2. jcsd
  3. Apr 24, 2007 #2
    I would integrate the curve 'r' wrt t, taking the limit to be 0 to 2pi.
  4. Apr 24, 2007 #3
    This is easily done (I think) using polar coordinates.
    If r(t) is the distance from the origin to the curve,
    and if t is the polar angle around the origin,
    then the surface is given by

    Integral[r(t)²/2 dt,{t,tmin,tmax)]

    You have to be careful to decide the limits of integration,
    and therefore you need to make a correct graphic of this function to understand the shape you need to evaluate.
    There may be some calucations to perform ...


    On the lhs, you wrote r in bold.
    Be careful, r is the distance in polar coordinates, it is not a vector and should not be written in bold.
    Last edited: Apr 24, 2007
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