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Finding the Arc Length of a Polar Function

  1. Mar 22, 2010 #1
    Hi, I've been having some issues in solving this problem.

    1. The problem statement, all variables and given/known data
    Find the arc length of r=2/(1-cosθ) from π/2 to π

    2. Relevant equations
    L =(integrate) sqrt(r2+(dr/dθ)2)dθ

    3. The attempt at a solution

    I found (dr/dθ) = (-2sinθ)/(1-cosθ)2

    so (dr/dθ)2 = (4sin2θ)/(1-cosθ)4

    Then r2 = 4/(1-cosθ)2

    both have a factor of 4/(1-cosθ)2, so I pulled that outside the sqrt to get

    L=(integrate) 2/(1-cosθ) * sqrt(1+(sin2θ)/(1-cosθ)2)

    then I multiplied the 1 by (1-cosθ)2/(1-cosθ)2 to give common denominators.

    After multiplying it out, the numerator of the fraction was 1-2cosθ+cos2θ+sin2θ, so I got rid of the sin and cos and added a 1 to get 2 - 2cosθ

    I pulled out a factor of sqrt2 and ended up with:

    L = (integrate) 2sqrt2/(1-cosθ) * sqrt(1/(1-cosθ))
    L= (integrate) 2sqrt2*(1-cosθ)-3/2

    This is where I got stuck. I can't think of any way to integrate that problem using any of the means we have gone over so far.

    Some direction in which way to go would be extremely helpful, thanks in advance.
    Last edited: Mar 22, 2010
  2. jcsd
  3. Mar 22, 2010 #2


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    Can you think of a way to express 1-cos(theta) as the square of something using a half-angle formula?
  4. Mar 23, 2010 #3
    Aha, thank you so much. That was just the push I needed!
  5. Mar 23, 2010 #4
    Hmm, that did allow me to solve the integration (at least, I think I did it properly). But now, when I go to plug in my evaluation, I'm getting sec(π/2) which is doesn't work out.

    Using the half angle, I got (1-cosθ)=2cos2(θ/2).

    Plugging that in canceled out my 2sqrt2, and made the integration into sec3(θ/2)

    Solving that gave me: sec(θ/2)tan(θ/2) + ln|sec(θ/2)+tan(θ/2)|

    Now in evaluating from π to π/2, plugging in for π is not a finite number. Did I go wrong in my integration of sec3?
  6. Mar 23, 2010 #5


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    1-cosθ is not equal to 2cos2(θ/2); it's equal to 2sin2(θ/2).
  7. Mar 23, 2010 #6
    Thanks! That's what I get for trying to do my homework far too late at night and without my list of identities nearby ~_~
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