Computing Arc Length and Cannot Solve the Integral

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SUMMARY

The discussion focuses on calculating the arc length of the polar curve defined by the cardioid equation r = 1 + cos(θ). The integral for the arc length is given by L = ∫αβ√[f(θ)² + f'(θ)²] dθ, where f(θ)² simplifies to cos²(θ) + 2cos(θ) + 1. The user struggles to solve the integral ∫αβ√[2cos(θ) + 2] dθ, indicating difficulties with techniques such as u-substitution and integration by parts.

PREREQUISITES
  • Understanding of polar coordinates and curves
  • Knowledge of integral calculus, specifically arc length formulas
  • Familiarity with trigonometric identities
  • Experience with integration techniques such as u-substitution and integration by parts
NEXT STEPS
  • Review trigonometric identities related to cosine functions
  • Practice solving integrals involving square roots and trigonometric functions
  • Study the derivation and application of the arc length formula in polar coordinates
  • Explore advanced integration techniques for complex integrals
USEFUL FOR

Students studying calculus, particularly those focusing on polar coordinates and arc length calculations, as well as educators seeking to reinforce integral calculus concepts.

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Homework Statement



Find the length of the polar curve.

Complete the cardioid r=1+cosθ

Homework Equations



L=∫αβ√[f(θ)2+f'(θ)2] dθ

The Attempt at a Solution



Given f(θ)2 is equal to cos2θ+2cosθ+1

and f'(θ)2=sin2θ

I arrive at the integral ∫αβ√[2cosθ+2] dθ which I cannot for the life of me solve for. It's been a while since my calc II class so I flipped through my textbook to see if I could find a solution. I cannot do u substitution as far as I can tell nor integration by parts and I couldn't find anything similar to the form above in the table of integrals.
 
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