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Arc length problem with a thorny integration

  1. Aug 12, 2012 #1
    1. The problem statement, all variables and given/known data

    So, the question gives a particle travelling over a path [itex]\gamma[/itex], and I need the arc length.

    2. Relevant equations

    The path is [itex]\gamma(t) : [1,4] \to ℝ^3, t \mapsto (t^2/2, t, ln(2t))[/itex].

    We want the arc length over [itex]1 \le t \le 4[/itex].

    3. The attempt at a solution

    First, the speed differential: [itex] ds = \left\| \gamma'(t) \right\| dt = \sqrt{t^2 + 1 + 1 /t^2} dt [/itex]

    Now, the arc length. [itex]\ell = \int_\gamma ds = \int_1^4 \sqrt{t^2 + 1 + 1 /t^2}dt[/itex].

    But that's where the fun ends. I've tried a bunch of trig substitutions (e.g. [itex]t=\tan u[/itex]), to no avail.

    I also tried Wolfram online integrator, which returned a mess of symbols -- this problem should have a (reasonably) simple analytic solution.

    Any ideas, anyone? I'd really appreciate any help!
  2. jcsd
  3. Aug 12, 2012 #2


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    Didn't you notice that [itex]t^2+ 1+ 1/t^2= t^2+ 2+ 1/t^2- 1= (t+ 1/t)^2- 1[/itex]?
  4. Aug 12, 2012 #3
    I had, but didn't realise it would help. I'll play around and see what I can come up with. :)

    Thankyou for the quick response!
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