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Thread: Help Me Immediately

  1. #1
    Apr 2009

    Help Me Immediately

    I just completed all paractice test except these 2 question.
    I try my best but I can not do anything.
    Help me, please.

    1. Find the area of the region that is enclosed by the rose r = 4cos3Θ
    2. Find the area of the region that is inside the circle r = 3sinΘ and outside the cardioids r = 1+sinΘ
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  2. #2
    Eater of Worlds
    galactus's Avatar
    Jul 2006
    Chaneysville, PA
    This is a 3 petal rose. To find the limits of integration, we can set

    $\displaystyle 4cos(3t)=0$ and find that $\displaystyle t=\frac{\pi}{6}$

    This gives the area of half a petal in the first quadrant. So we multiply by 2.

    $\displaystyle \int_{0}^{\frac{\pi}{6}}[4cos(3t)]^{2}dt$

    For the other one, we can find the region in the first quadrant and multiply by 2:

    Where they intersect can be found by $\displaystyle 3sin(t)=1+sin(t)$

    As before, $\displaystyle t=\frac{\pi}{6}$

    $\displaystyle \int_{\frac{\pi}{6}}^{\frac{\pi}{2}}\left[(3sin(t))^{2}-(1+sin(t))^{2}\right]dt$

    $\displaystyle \int_{\frac{\pi}{6}}^{\frac{\pi}{2}}\left[-4cos(2t)-2sin(t)+3\right]dt$
    Attached Thumbnails Attached Thumbnails Help Me Immediately-cardiod.jpg   Help Me Immediately-rose3.jpg  
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