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Calcularing area vector using line integral

  1. Sep 15, 2015 #1
    1. The problem statement, all variables and given/known data
    A closed curve C is described by the following equations in a Cartesian coordinate system:
    gif.gif
    gif.gif
    gif.gif
    where the parameter t runs monotonically from 0 to 2π, thus defining the direction of C. Calculate the area vector of the planar region enclosed by C, using the formula:
    gif.gif

    2. The attempt at a solution
    I'm mostly having trouble defining what gif.gif is as a physical quantity. I think it is the distance to an incremented point along the curve such that gif.gif it the area of the equilateral shape formed by the vectors and that half the integral of that gives the area but i'm suck here:
    gif.gif
    Than I can evaluate it as:
    gif.gif
    and this would give a result that is only in the z direction which dimensionally makes sense
     
  2. jcsd
  3. Sep 15, 2015 #2

    Ray Vickson

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    If
    [tex] \vec{r} = x(t) \, {\bf i} + y(t) \, {\bf j} + z(t)\, {\bf k} [/tex]
    how would you compute ##d \vec{r}## in terms of ##dt##?
     
  4. Sep 16, 2015 #3
    My first instinct was just to derive each of them with respect to t like such
    20y%28t%29%5Chat%7By%7D+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20t%7D%20z%28t%29%5Chat%7Bz%7D.gif
    gif.gif
    Is that right?
     
  5. Sep 16, 2015 #4
  6. Sep 16, 2015 #5

    SteamKing

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    Yes, go ahead and finish the evaluation of the integral ...
     
  7. Sep 16, 2015 #6
    So http://www.sciweavers.org/upload/Tex2Img_1442441349/eqn.png [Broken] (by trig identities)
    and so the integral is
    http://www.sciweavers.org/upload/Tex2Img_1442441225/eqn.png [Broken]
     
    Last edited by a moderator: May 7, 2017
  8. Sep 16, 2015 #7

    SteamKing

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    There's a small mistake in your integration. You seem to have omitted integrating the constant (3/8) in your trig identity expression
     
    Last edited by a moderator: May 7, 2017
  9. Sep 16, 2015 #8
    I was hoping you wouldn't notice that, I was just too lazy to retype it into LaTex, but yes, it was included in my calculations (I appreciate your thoroughness though!)
     
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