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Tricky integral from calc 3 Arc Length question

  1. Feb 19, 2015 #1

    RJLiberator

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    1. The problem statement, all variables and given/known data
    Find the arc length of:

    r(t)=<e^t, e^(-t),sqrt(2)*t>
    from 0 to ln(2)

    2. Relevant equations
    L=integral from a to b of the magnitude of r'(t)

    3. The attempt at a solution

    Okay, this was an Exam question, the one exam question that I could not get on our Calc 3 exam. This breaks down into an integral. The answer is 3/2 and there seems to be an obvious trick that I am missing.

    r'(t)=<e^t, -e^(-t), sqrt(2)>
    Take the magnitude
    sqrt((e^t)^2+(-e^-t)^2+2)
    Take the integral of that from 0 to ln(2).

    There is some trick here that I simply could not get. I spent an hour on this problem during the exam :p.
     
  2. jcsd
  3. Feb 19, 2015 #2

    LCKurtz

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    Write that as$$
    \sqrt{e^{2t} + 2 +e^{-2t}}$$and note that the quantity under the square root sign is a perfect square.
     
  4. Feb 19, 2015 #3

    RJLiberator

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    I'm still not getting it. I've searched google for 'perfect square' and I see what it means, but how is it visible in this instance?
     
  5. Feb 19, 2015 #4

    Dick

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    Guess a factorization. If that's equal to (a+b)^2 what do you think a and b might be? Here's a hint. (e^x)^2=e^(2x). After you've guessed, check it.
     
    Last edited: Feb 19, 2015
  6. Feb 20, 2015 #5

    RJLiberator

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    Ugh, I sat here and thought about it for 30 minutes, but something isn't sticking out to me.

    I know it must be in form (a+b)^2 so that the square root will cancel and the problem will be easily solvable.
    Hm.
    so we have: (e^(2x)+2+e^(-2x)).
    The negative sign is throwing me off greatly.
     
  7. Feb 20, 2015 #6

    HallsofIvy

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    [itex]e^{2x}= (e^x)^2[/itex]. [itex]e^{-2x}[/itex] is equal to [itex](e^{-x})^2[/itex].
     
  8. Feb 20, 2015 #7

    RJLiberator

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    Oh... I just got it, it clicked to perfection:
    (e^x+e^(-x))^2

    This way the e^(-x)*e^x cancel out to create 1+1=2.
    A perfect square.

    And then taking out the square root you simply are left with the integral of e^x+e^(-x).
     
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