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Compute the length of the cardioid below the x-axis?

  1. Mar 18, 2012 #1
    [tex]r = 2sin(\theta)-2[/tex]

    First we find x(θ), y(θ)

    [tex]x(\theta) = rcos(\theta)[/tex]
    [tex]y(\theta) = rsin(\theta)[/tex]

    Then we find x'(θ) and y'(θ) to use the formula:

    [tex]L = \int_\alpha^β \sqrt{x'(\theta)^2 + y'(\theta)^2} d\theta[/tex]

    My problem is that I don't know how to get the limits of integration. The answer key says that they are from 0 to π, but I would have guessed π to 2π, since that represents everything below the x-axis? Any help would be appreciated.

    Thanks, guys!
     
  2. jcsd
  3. Mar 18, 2012 #2

    LCKurtz

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    Why guess? It's going to cross the x axis when y = 0. What values of ##\theta## do that? Plot the graph for additional information on what direction it is going when it crosses the x axis.
     
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