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Math Help - Polar graph integration

  1. #1
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    Polar graph integration

    Question: What is the area bounded by one petal of the graph r = (1 - Cos(3theta))?


    From plotting this in the polar and Cartesian plane I gathered that one petal occurs on the interval (0, 2pi/3).

    Therefore the area would be 1/2 * (Integral from 0 to 2pi/3 of (1-Cos(3theta))^2 which gives me approximately 4.72. However according to the online quiz this is not the correct answer. What am I doing wrong here? I even double checked with an online integration calculator and got the same value.
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Polar graph integration

    I get:

    \frac{1}{2}\int_0^{\frac{2\pi}{3}} (1-\cos(3\theta))^2\,d\theta=2\left[\frac{3}{8}x-\frac{1}{6}\sin(3\theta)+\frac{1}{48}\sin(6\theta) \right]_0^{\frac{2\pi}{3}}=

    2\left(\frac{3}{8}\cdot\frac{2\pi}{3} \right)=\frac{\pi}{2}
    Thanks from x3bnm
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