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Integrating to find the length of a Cardiot Curve

  1. Nov 28, 2009 #1
    1. The problem statement, all variables and given/known data
    find the length of the cardiot r = 1+cos(Θ)

    I'm going to use { as the integral sign
    all integrals are definite between 0 and Pi

    2. Relevant equations
    L = 2 {sqrt[(r^2(Θ))+(dr/dΘ)^2]dΘ

    3. The attempt at a solution
    L=2* {sqrt[2+2cos(Θ)] dΘ

    I'm having a really hard time trying to integrate sqrt[2+2cos(Θ)] dΘ and was hoping someone could explain to me how you integrate that.

  2. jcsd
  3. Nov 28, 2009 #2
    Personally, I would start by doing

    \int_0^\pi \sqrt{2 + 2\cos\theta} \, d\theta
    = \sqrt{2}\int_0^\pi \sqrt{1 + \cos\theta}\, d\theta.

    It's totally not necessary, but it sort of simplifies things. Then you want to use the identity
    \frac{1 + \cos\theta}{2} = \cos^2\frac{\theta}{2}

    Multiply both sides of that identity by 2, then you'll have something a little nicer to work with..
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