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Using Green's Theorem

  1. Jul 27, 2010 #1
    1. The problem statement, all variables and given/known data
    Use a line integral to find the area of the region enclosed by astroid
    x = acos3[tex]\phi[/tex]
    y = asin3[tex]\phi[/tex]

    0 [tex]\leq \phi \leq 2\pi[/tex]

    2. Relevant equations

    I used Green's Theorem:

    [tex] \oint_C xdy - ydx[/tex]

    3. The attempt at a solution
    I solved for dx and dy from my parametric equations. I then plugged in x, y, dx, and dy into the integral to solve for the area.

    After simplifying, I came out with:

    [tex] \frac {3a}{2} \int_0^{2\pi} cos^2\phi sin^2\phi d\phi[/tex]

    Now in order to solve this, I used a half angle formula, [tex]cos\phi sin\phi = (\frac{1}{2}sin2\phi)^2 = \frac {1}{4}sin^2 2\phi [/tex]

    Which then I used a different angle formula to get:[tex] \frac{1}{8}(1-cos4\phi)[/tex]

    Am I on the right track? I would then integrate to solve...

    The latex on my computer isnt working, but hopefully its working on everyone else's?
     
  2. jcsd
  3. Jul 28, 2010 #2

    hunt_mat

    User Avatar
    Homework Helper

    I get the coefficent outside the integral to be 3a^{2} but apart from that I see nothing wrong with this working.
     
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