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Determine the length of the curve sin(x)

  1. Mar 21, 2012 #1
    What is the measure of the sin(x) wave for x=0 to 2∏?
  2. jcsd
  3. Mar 21, 2012 #2


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    It's [tex]\int^{2\pi}_0\sqrt{\cos(x)^2+1} dx[/tex]
  4. Mar 21, 2012 #3
    That's what I got. Would one need a table of integrals to determine its numerical value?
  5. Mar 21, 2012 #4


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    Pretty straight forward, isn't it? Considering the other problems you have posted on here, you should be able to do this.

    The length of the graph of y= f(x), from x= a to x= b, is given by
    [tex]\int_{x=a}^b \sqrt{1+ f'(x)^2}dx[/tex]
    With y= f(x)= sin(x), f'(x)= cos(x) so that becomes
    [tex]\int_{x=0}^{2\pi} \sqrt{1+ cos^2(x)}dx[/tex]
    However, that looks to me like a version of an elliptical integral which cannot be done in terms of elementary functions.

    Hey, no fair posting while I'm typing!
  6. Mar 21, 2012 #5

    D H

    Staff: Mentor

    Yep. It's 4√2E(1/2), where E(x) is the complete elliptical integral of the second kind.
  7. Mar 24, 2012 #6


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    to get a numerical value try numerical integration, simpson's rule? etc...

    this is no worse than finding the area under the curve from 0 to 1. i.e. both are approximations.

    (nobody knows what cos(1) is.)
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