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Line Integral

  1. Jan 29, 2007 #1
    1. The problem statement, all variables and given/known data
    This is my problem:
    Compute the following three line integrals directly around the boundary C of the part R of the interior ellipse (x^2/a^2)+(y^2/b^2)=1 where a>0 and b>0 that lies in the first quadrant:
    (a) integral(xdy-ydx)
    (b) integral((x^2)dy)
    (c) integral((y^2)dx)

    2. Relevant equations
    I used parametrisation (x=acost and y=bsint) for the arc of the ellipse.
    C is the curve r=(acost)i + (bsint)j (0 less than or equal to t less than or equal to pi).

    3. The attempt at a solution
    (a) integral(xdy-ydx)=integral from 0 to pi((acost)(bcost)dt)- integral from 0 to pi((bsint)(-asint)dt)=(ab)pi/2-(-ab)(pi)/2=ab(pi)

    (b) integral((x^2)dy)=integral from 0 to pi((acost)(acost)(bcost)dt)=0.

    (c) integral((y^2)dx)=integral from 0 to pi((bsint)(bsint)(-asint)dt)=-(4/3)a(b^2)

    Could anyone check these and see if they are right?
  2. jcsd
  3. Jan 29, 2007 #2
    Actually, should all the bounds be from 0 to pi/2 instead of 0 to pi (since I am looking at only the first quadrant)?
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