Determine the length of the curve sin(x)

  1. What is the measure of the sin(x) wave for x=0 to 2∏?
     
  2. jcsd
  3. disregardthat

    disregardthat 1,811
    Science Advisor

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

    HallsofIvy 40,385
    Staff Emeritus
    Science Advisor

    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. D H

    Staff: Mentor

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

    mathwonk 9,690
    Science Advisor
    Homework Helper

    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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