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Math Help - areas and lengths in polar coordinates

  1. #1
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    areas and lengths in polar coordinates

    a few questions

    1.) find the area enclosed by one loop of the curve r= cos 4 theta

    2.) find the area enclosed by one loop of the curve r= 2sin(5theta)

    3.) find the area of the region that lies inside the first curve and outside the second curve

    r= 5cos(theta)
    r= 2+cos(theta)

    4.) find the area inside the larger loop and outside the smaller loop of the limacon r= root 3/2 (only the 3 in root) + cos(theta)
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  2. #2
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    Hello, Gauss's law!

    These are basically of the same type: . A \;=\;\tfrac{1}{2}\int^{\beta}_{\alpha} r^2\,d\theta
    Exactly where is your difficulty?


    1) Find the area enclosed by one loop of the curve: .  r\:=\:\cos4\theta
    This is an 8-leaf rose curve with one leaf on the 0-line.

    To find the limits, solve r = 0.
    . . \cos4\theta \:=\:0 \quad\Rightarrow\quad 4\theta \:=\:\pm\tfrac{\pi}{2} \quad\Rightarrow\quad \theta \:=\:\pm\tfrac{\pi}{8}

    Due to its symmetry, we can integrate from 0 to \tfrac{\pi}{8} and multiply by 2.


    We have: . A \;=\;2\times\tfrac{1}{2}\!\int^{\frac{\pi}{8}}_0 \cos^2\!4\theta\,d\theta \;=\;\tfrac{1}{2}\!\int^{\frac{\pi}{8}}_0 (1 + \cos8\theta)\,d\theta


    Can you finish it?


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  3. #3
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    The step you stopped on is the step I have problems with actually.
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