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Math Help - Line Integral

  1. #1
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    Line Integral

    If F(x)= (3xy, 2x-y), find

    <br />
\int\limits_\gamma F(x)\,dx<br />

    where \gamma is the directed path from (0,0) to (1,1) along the graph of the vector equation:

    x = (\sin t, 2t/\pi),  (0 \leq t \leq \pi/2)

    __________________________________________________ _____

    Here's what I did:

    I recognized that the integral is not independent of path... So set up:

    \int M dx + N dy

    which gives me the sum of 4 integral terms... I evaluated these up to the following step so somebody could double check:

    6/\pi [\pi/2 -1] - 6/\pi[\pi/3 - 7/9] + 4/\pi - 1/2


    Now does this answer seem correct to you?
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  2. #2
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    I do NOT get the sum of four terms, I get the sum of three terms:

    On the path x= sin t, y= 2t/\pi, dx= cos t dt and dy= 2/\pi dt.

    3x^2y= 6 tsin^2(t)/\pi and 2x- y= 2sin(t)- 2t/\pi

    The integral becomes 6 \int_{t= 0}^{\pi/2} t sin^2 t cos(t) \, dt+ 4\pi\int_{t=0}^{\pi/2}sin(t) \, dt- 4\pi\int_{t=0}^{\pi/2} t\,  dt.

    Use integration by parts to do the first integral.
    Last edited by mr fantastic; July 3rd 2010 at 05:02 PM. Reason: Fixed some latex.
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  3. #3
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    Thank you HallsofIvy

    Quote Originally Posted by HallsofIvy View Post
    The integral becomes 6 nt_{t= 0}^{\pi/2} t sin^2 t cos(t)dt+ 4\pi\int_{t=0}^{\pi/2}sin(t)dt- 4\pi\int_{t=0}^{\pi/2} t dt.

    Use integration by parts to do the first integral.
    But I think your integral is incorrect. Did you mean:

    6/\pi \int_{t= 0}^{\pi/2} t sin^2 t cos(t)dt+  4/\pi\int_{t=0}^{\pi/2}sin(t)dt- 4/\pi^2\int_{t=0}^{\pi/2} t dt

    If this is correct, how would you integrate the first one by parts?
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  4. #4
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    Quote Originally Posted by keysar7 View Post
    Thank you HallsofIvy



    But I think your integral is incorrect. Did you mean:

    6/\pi \int_{t= 0}^{\pi/2} t sin^2 t cos(t)dt+ 4/\pi\int_{t=0}^{\pi/2}sin(t)dt- 4/\pi^2\int_{t=0}^{\pi/2} t dt

    If this is correct, how would you integrate the first one by parts?
    Yes, there were a couple of typos in HoI's reply which you have spotted and fixed.

    See integrate x &#40;Sin&#91;x&#93;&#41;&#94;2 Cos&#91;x&#93; - Wolfram|Alpha and click on Show steps.
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