Arc Length of Ellipse, Hard Integral

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SUMMARY

The discussion focuses on finding the arc length of the ellipse defined by the equation r² = x² + (y/β)², where r is the radius and β is the dilation constant. Participants concluded that the integral required to solve this problem involves elliptic integrals, which are known to be non-expressible in terms of elementary functions. The conversation emphasizes the complexity of elliptic integrals and their significance in advanced mathematical analysis.

PREREQUISITES
  • Understanding of elliptic integrals
  • Familiarity with integral calculus
  • Knowledge of conic sections, specifically ellipses
  • Basic concepts of dilation in geometry
NEXT STEPS
  • Study the properties and applications of elliptic integrals
  • Learn techniques for numerical integration of complex functions
  • Explore the theory behind conic sections and their equations
  • Investigate advanced calculus topics related to arc length calculations
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Mathematics students, educators, and researchers interested in advanced calculus, particularly those dealing with elliptic integrals and arc length problems in geometry.

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Homework Statement


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

Homework Equations


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



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
 
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Check out the term elliptic integrals. You will find a fully developed theory behind this whole area.
 
OldEngr63 said:
Check out the term elliptic integrals. You will find a fully developed theory behind this whole area.
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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