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Math Help - Help Me Immediately

  1. #1
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    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
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    This is a 3 petal rose. To find the limits of integration, we can set

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

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

    \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 3sin(t)=1+sin(t)

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

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

    \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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