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Computing The Arclength Function

  1. Nov 18, 2008 #1
    1. The problem statement, all variables and given/known data

    Consider the curve r = (e^−2 t cos(3 t), e^−2 t sin(3 t), e^−2 t) .

    Compute the arclength function s(t) : (with initial point t=0 ).



    3. The attempt at a solution

    r'(t) = <-2e^-2t*cos(3t) + e^-2t*-3sin(3t), -2e^-2t*sin(3t) + e^-2t*cos(3t), -2e^-2t>

    Then what, do I find the length of that derivative?
    Then take the integral of 0 to t?
    I dunno.
     
  2. jcsd
  3. Nov 18, 2008 #2
    Hi Withthemotive,

    A minor point: the e^(-2t)*cos(3t) term in your second component is missing a factor of 3. Also, be sure to use plenty of parentheses to remove any ambiguity in the future!

    That's right, but your derivative will need to be with respect to a dummy variable, say u:
    [tex]s(t) = \int_0^t ||\text{r}'(u)|| \, \text{du}[/tex].
     
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