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Line Integrals

  1. Jan 20, 2012 #1
    1. The problem statement, all variables and given/known data
    Calculate the folowing directly and with greens theorem


    2. Relevant equations

    [itex] \int (x-y) dx + (x+y) dy[/itex]

    C= x^2+y^2=4

    3. The attempt at a solution

    Directly

    [itex]x= r cos \theta, y=r sin \theta, r^2=4, dx = -r sin \theta d \theta, dy= r cos \theta d \theta[/itex]

    Substituting I get

    [itex] \displaystyle \int_0^{2 \pi} (-r^2 sin \theta cos \theta +r^2 sin^2 \theta) d \theta+(r^2 cos^2 \theta +r^2 sin \theta cos \theta) d \theta[/itex]

    [itex]=4 \int_0^{2 \pi} d \theta= 8 \pi[/itex]

    Greens theorem

    [itex] \displaystyle \int \int_R (G_x -F_y)dA= \int_0^{2 \pi}\int_0^2 2 r dr d \theta = 2 \pi[/itex].....? I cant spot the error!
     
  2. jcsd
  3. Jan 20, 2012 #2

    LCKurtz

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    I can't spot the error either because you didn't show your [incorrect] work to get ##2\pi##.
     
  4. Jan 21, 2012 #3
    I have spotted it this morning. Just used wrong limits in calculation although shown correctly above. Late night concentration I guess.

    Thanks LCKurtz
     
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