Trig function appears awkward to integrate

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SUMMARY

The discussion centers on integrating the function √(sin²θ + k cos²θ) to determine the length of the locus described by a garage door as it is raised. The user, an engineer, initially attempted various methods but ultimately resorted to numerical integration, which yielded results consistent with real-world observations. The conversation highlights the potential use of elliptic functions for a more elegant solution, specifically referencing the Elliptic Integral of the Second Kind as a valuable resource.

PREREQUISITES
  • Understanding of trigonometric functions and their properties
  • Familiarity with numerical integration techniques
  • Basic knowledge of elliptic integrals
  • Experience with mathematical modeling in engineering contexts
NEXT STEPS
  • Study the properties and applications of elliptic integrals, particularly the Second Kind
  • Explore advanced numerical integration methods for complex functions
  • Learn about trigonometric substitutions in integral calculus
  • Investigate practical applications of elliptic functions in engineering
USEFUL FOR

Engineers, mathematicians, and students involved in calculus or numerical methods, particularly those interested in integrating complex trigonometric functions.

Edmundb
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I hope this isn't too simple for the maths forum.

I'm trying to find the length of the locus described by the bottom of a garage door as it is raised. It's all fairly straightforward until I have to integrate a function of the form

√ (sin2θ + k cos2θ)

k does not depend on θ. I tried several approaches to the problem and always came back to the above function.

In the end I integrated numerically and got my number which checks with reality. There's probably some nifty substitution but just I can't find it.
 
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Thank you very much indeed. That looks like the clue I need. (As you've probably gathered I'm an Engineer and not a mathematician.)
 

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