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U-Substitution with du=0

  1. Nov 13, 2011 #1
    1. The problem statement, all variables and given/known data

    I am trying to prove that the length of a helix can be represented by [itex]2\pi=\sqrt{a^2+b^2}[/itex]

    2. Relevant equations



    3. The attempt at a solution

    I have the following so far:

    If the helix can be represented by [itex]h(t)=a\cdot cos(t)+a\cdot sin(t)+b(t)[/itex]

    Then the length is:
    [tex]\int_{0}^{2\pi}\sqrt{(-a\cdot sin(t))^2+(a\cdot cos(t))^2+b^2}\;\: dt[/tex]

    My problem comes when integrating this. If I use the stuff in the root as u and do u-substitution, then du equals 0dt:

    [tex]u=a^2sin^(t)+a^2cos^2(t)+b^2[/tex]
    [tex]du=(a^2sin(2t)-a^2sin(2t))dt=0dt[/tex]

    My logic fails me when figuring out how to continue from there. I need to somehow represent 1dt. How do I do this?

    Help would be awesome!
     
  2. jcsd
  3. Nov 13, 2011 #2

    Dick

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    Science Advisor
    Homework Helper

    If your du comes out to be zero, then u must be a constant. What constant is it? Use some trig to simplify your u.
     
  4. Nov 13, 2011 #3

    gb7nash

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    Homework Helper

    You don't need u-substitution for this problem. Here's why:

    Before you try taking the integral, inside the square root, you have a2sin2t + a2cos2t. Factor the a2 out of both of them. What happens?
     
  5. Nov 13, 2011 #4
    Yep, was definitely over-thinking it. Thanks guys :)
     
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