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Finding line integrals - please help!

  1. Apr 21, 2009 #1
    Finding line integrals -- please help!

    Given
    F = y/(x^2 + y^2) i - x / (x^2 + y^2) j

    Find the line integral of the tangential component of F from (-1,0) to (0,1) to (1,1) to (1,0) (assuming F is NOT path independent).

    ---

    I tried parameterizing each of the three paths using the formula
    r(t) = (1-t)r_0 + tr_1

    ---

    I've been racking my brain on this for hours. The book says the answer is pi. My teacher says it isn't.

    For path 1:
    x = -1 + t, y = t
    => dx = dt = dy

    For path 2:
    x = t, y = 1
    => dy = 0

    For path 3:
    x = 1, y = 1-t
    => dx = 0

    I then substituted these values in the original equation for F and integrated from t = 0 to t = 1 for each step. I get

    [tan^-1 (1) - tan^-1 (1)] + [tan^-1 (1) - tan^-1 (0)] + [tan^-1 (0) - tan^-1 (1)] = 0 + pi/4 - pi/4 = 0

    Perhaps I take one of the tan^-1 (0) to be pi instead? What am I doing wrong? Please help!
     
    Last edited: Apr 22, 2009
  2. jcsd
  3. Apr 22, 2009 #2
    Re: Finding line integrals -- please help!

    I did the first path and got pi/2, not 0. What integral did you get?

    Remember, the integral is int_0^1 F(r(t)) * dr = int_0^1 F(r(t)) * dr/dt dt, where * denotes the dot product. For this path, since r = (-1+t, t), dr/dt is (1,1). After working it out, it came down to int_0^1 1/((-1+t)^2 + t^2) dt which you solve by completing the square on the bottom so you'll end up with an tan^-1. Hopefully you can read this-- unfortunately TeX isn't working.
     
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