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How long is a piece of sinosoidal string?

  1. Nov 6, 2007 #1
    I am trying to calculate the arc length of a sine wave.

    Using [tex]s=\int_{}^{}\sqrt[]{1 + {(\frac{dy}{dx})}^{2}}dx[/tex]

    if y = sinx, dy/dx = cosx

    So the integral simplyfies to [tex]s=\int_{}^{}\sqrt[]{1 + {cos}^{2}(x)}dx[/tex]

    However I do not know any integration technique (ie. substitution, by parts, etc..) with which I can calculate this integral analytically.

    If you can think of any other way of going about this, any help would be greatly appreciated.

  2. jcsd
  3. Nov 6, 2007 #2


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    Mathematica calls this function EllipticE (performing the integral from 0 to 2п gives [itex]4\sqrt{2}\mathtt{EllipticE}(1/2) \approx 7.6404[/itex]), so I doubt there is a more elementary answer (like [itex]\pi/2[/itex] or [itex]\operatorname{arcsinh}(-1)[/itex]).
  4. Nov 6, 2007 #3
    Right, there is no elementary answer
  5. Nov 6, 2007 #4
    I have to admit I am a little disappointed. I thought there might be a way of performing the integral by pure analytical means.
    But thank you very much for your responses.
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