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Math Help - Arc length of a curve represented by vector function

  1. #1
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    Arc length of a curve represented by vector function

    I'm having some trouble doing this question where i'm supposed to find the arc length of a curve represented by a vector function.

    Question is: Find the arc length of the curve.

    r(t) = (\sqrt(2)*t) i + (e^t)j + 1/(e^t)k over the interval 0<=t<=1

    So since arc length = int(0,1)[|r`(t)|] i calculate r`(t):
    r`(t) = \sqrt(2)i + (e^t)j + -1/(e^t)k
    and
    |r`(t)| = \sqrt(2 + (e^t)^2 + -1/(e^t)^2 )
    and this is where i get stuck. I must have to plug something into |r`(t)| equation above for t to calculate |r`(t)| and then plug it into the arc length equation. How am i supposed to integrate |r`(t)| as it is shown in the above form?
    Last edited by mkelly09; November 28th 2009 at 10:24 AM.
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  2. #2
    Senior Member
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    Quick correction:

    |\mathbf{r'}(t)|=\sqrt{2+(e^t)^2+\frac{1}{(e^t)^2}  }.

    The integral

    \int_0^1\sqrt{2+e^{2t}+e^{-2t}}\,dt

    may be found by substitution. Hint: let t=\frac{1}{2}\ln u.
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