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Math Help - Line integral question

  1. #1
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    Line integral question

    The question
    For the vector field F (x, y, z) = (x^2 + y^2 + z^2)i - zj + (y + 1)k calculate \int_C{F.dr} where the path C from (0, 0, -1) to (0, 0, 1) is the circle y^2 + z^2 = 1 in the plane x = 0, the direction of motion being anti-clockwise when viewed from the positive x-axis.

    My attempt
    Let z = sin(t), y = cos(t)
    C(t) = (0, cos(t), sin(t))
    C'(t) = (0, -sin(t), cos(t))
    F(C(t)) = (cos^2(t) + sin^2(t))i - sin(t)j + (cos(t) + 1)k
    = i - sin(t)j + (cos(t) + 1)k

    (1, -sin(t), cos(t) + 1).(0, -sin(t), cos(t))
    = 1 + cos(t)

    \int_{-1 C}^{1}{1 + cos(t) \ dt}

    = 2 + 2sin(1)

    However the solution is 2 + \pi.

    Where have I gone wrong? Thanks.
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  2. #2
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    With the parametric equations you give, t goes from 0 to 2\pi (around the full circle), not -1 to 1.
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  3. #3
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    Ahh, I see. So it wouldn't be -\frac{\pi}{2} to \frac{3\pi}{2}? Or it doesn't matter?
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  4. #4
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    Hmm, now I'm getting 2\pi, which is also wrong. :/
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  5. #5
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    I can't see where I'm going wrong, is the rest of the working correct?
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  6. #6
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    the limits of your integration should be from -\pi/2 to \pi/2

    (you are only tracing out a semi-circle in the yz-plane if C goes from (0,0,-1) to (0,0,1) counter-clockwise along the circle).
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  7. #7
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    Thank you. I was a bit confused when HallsofIvy said 0 to 2pi.
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  8. #8
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    Sorry, I misread the problem, thinking you were integrating around the entire circle. I wondered where you got -1 and 1 from!
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