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Math Help - Polar Curves stuff

  1. #1
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    Polar Curves stuff

    find the area of the region inside one loop of the lemniscate r^2=4sin2theta
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  2. #2
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    Hello, leinadwerdna!

    Find the area of the region inside one loop of the lemniscate: r^2\:=\:4\sin2\theta
    We really have very little do . . .

    Formula: . A \;=\;\tfrac{1}{2}\int^{\beta}_{\alpha} r^2\,d\theta

    If you recognize the curve or can make a sketch, you see that
    . . it is a figure-8 tipped so the loops are in Quadrants 1 and 3.

    We already have r^2 . . . all we need are the limits.

    Set r = 0\!:\quad 4\sin2\theta \:=\:0\quad\Rightarrow\quad\sin2\theta \:=\:0 \quad\Rightarrow\quad 2\theta \:=\:0,\:\pi \quad\Rightarrow\quad \theta \:=\:0,\:\tfrac{\pi}{2}

    There! . . . A \;=\;\tfrac{1}{2}\int^{\frac{\pi}{2}}_0 4\sin2\theta\,d\theta

    Got it?

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