1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    User Avatar
    Science Advisor

    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!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook