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Integral Over CFds

  1. Sep 28, 2014 #1
    1. The problem statement, all variables and given/known data
    For F = (x^2+y)i + (y-x)j, calculate the integral over C of Fds for r = (t, t^2) where t goes from 0 to 1.

    2. Relevant equations


    3. The attempt at a solution
    I know the integral over C of fds is f*sqrt(r'(t)^2+r^2*theta'(t)^2) dt. But I have no theta in this question, is this the wrong integral?
     
  2. jcsd
  3. Sep 28, 2014 #2

    LCKurtz

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    Yes, that's the wrong integral. This problem has nothing to do with polar coordinates. You have ##\vec r(t) =\langle x(t),y(t)\rangle = \langle t,t^2\rangle##. Write the integral in terms of ##t##.
     
  4. Sep 28, 2014 #3
    So F would be (2t^2)i + (t^2 - t)j. And dr is just <1, 2t>dt. But how would I integrate with respect to ds if I end up with a dt?
     
  5. Sep 28, 2014 #4

    LCKurtz

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    Your integral is stated with an Fds. I don't think that is an arc length integral, but then you haven't told us what that notation means. F is a vector. What is ds? I would assume you mean ##\vec F\cdot d\vec s## which might otherwise be written ##\vec F\cdot \hat T~ds## or ##\vec F\cdot d\vec r##. In any case I expect you would evaluate it as ##\int\vec F\cdot \vec r'(t)~dt##.
     
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