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Elliptic line integral

  1. Sep 10, 2007 #1
    1. The problem statement, all variables and given/known data
    Let C be the ellipse with center (0,0), major axis of length 2a, and minor axis of length 2b. Evaluate [tex]\oint_C xdy - ydx[/tex].


    2. Relevant equations
    I solved this two ways. First I parameterized x and y as [tex]x=a \cos \theta[/tex] and similarly for y. I also applied Green's theorem, which yielded [tex]\oint_C xdy - ydx = 2 \int \int_D dA[/tex] where D is the area enclosed by C (ie an ellipse.) In both cases I got the answer [tex]2\pi a b[/tex].
    3. The attempt at a solution
    My only question is, the book I am using says the answer is [tex]\frac{\pi a b}{2}[/tex]. This is an ETS book and they don't usually have typos, especially when it's the answer key to a previously administered exam. What am I missing?
     
  2. jcsd
  3. Sep 10, 2007 #2

    Dick

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    The answer is pi*a*b/2. If a=b=r then it's a circle and the area is pi*r^2. So the contour is half that.
     
  4. Sep 11, 2007 #3
    I apologize for being so dense, but I'm still confused. A couple different books I have print the result

    [tex]\frac{1}{2}\oint_C -ydx + xdy = \iint_{R} dA = A[/tex]

    If the area of the ellipse is [tex]A=\pi a b[/tex] then I would think that the value of the line integral is [tex]2A[/tex].
     
  5. Sep 11, 2007 #4

    Dick

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    Sorry, yes, I think you are right. Don't know what I was thinking...
     
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