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Arc Length of Ellipse, Hard Integral

  1. Apr 16, 2012 #1
    1. The problem statement, all variables and given/known data
    Find the arc length of the ellipse or deformed circle. r^2=x^2+(y/β)^2

    r=radius
    β=dilation constant
    k="random" constant

    2. Relevant equations
    gif.latex?Arc%20Length=\int%20dx\sqrt{1+%28\frac{dy}{dx}%29^{2}}.gif


    3. The attempt at a solution
    I suspect it's impossible but I can't prove that.
    After working it out I got stuck with these integrals which I can not solve:

    gif.latex?\int%20\frac{csc%20\theta%20\cdot%20sec^{2}\theta}{\sqrt{1-k\cdot%20tan\theta%20}}dx.gif

    r^2}}.gif
     
  2. jcsd
  3. Apr 16, 2012 #2

    OldEngr63

    User Avatar
    Gold Member

    Check out the term elliptic integrals. You will find a fully developed theory behind this whole area.
     
  4. Apr 16, 2012 #3
    Thanks. I had googled some pages that hinted at that but this statement in wiki about "elliptic integrals" confirmed it: "In general, elliptic integrals cannot be expressed in terms of elementary functions."
     
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