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Math Help - Area in Polar Coordinates

  1. #1
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    Area in Polar Coordinates

    So, I've tried this problem about 10 times, and I'm not quite sure what I'm doing wrong.

    Find the area inside one leaf of the rose
    r= 4sin(5θ)

    I know the graph looks like a 5 leaf rose.
    I used the equation A= integral (1/2r^2)dθ

    Maybe I'm bounding my integral wrong?? I have it from 0 to pi/5
    Or maybe my antiderivative is wrong??? I have 4θ - (4/10)(sin10θ)

    Any help would be greatly appreciated! This question is really starting to frustrate me!
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  2. #2
    Eater of Worlds
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    As you done, you can use the limits 0 to Pi/5, then since there is symmetry, you can multiply by 5.

    \frac{5}{2}\int_{0}^{\frac{\pi}{5}}\left(4sin(5\th  eta)\right)^{2}d{\theta}=40\int_{0}^{\frac{\pi}{5}  }sin^{2}(5\theta)d{\theta}

    When integrating a sin^{2}({\theta}), a good identity to use is

    sin^{2}(\theta)=\frac{1}{2}(1-cos(2{\theta})).

    =20\int_{0}^{\frac{\pi}{5}}\left(1-cos(10\theta)\right)d{\theta}

    =20{\theta}-2sin(10{\theta})

    20(\frac{\pi}{5})-2sin(10(\frac{\pi}{5}))=4{\pi}
    Last edited by galactus; November 24th 2008 at 05:38 AM.
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