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Thread: Find the area of the specified region

  1. #1
    Super Member bigwave's Avatar
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    Find the area of the specified region

    $\textsf{Find the area of the specified region} $
    $\textsf{Shared by the circle $r=8$ and the cardioid $r=8(1+\sin{\theta})$}$
    $\textsf{Area of the Region $0 \le r_1(\theta) \le r_2(\theta)
    \, \, \alpha \le \theta \le \beta$}$
    \begin{align*}\displaystyle
    A&=\int_{\alpha}^{\beta}\frac{1}{2}r_2^2 \, d\theta
    +\int_{\alpha}^{\beta}\frac{1}{2}r_1^2 \, d\theta
    =\frac{1}{2} \int_{\alpha}^{\beta}(r_2^2+r_1^2 )\, d\theta
    \end{align*}
    $\textit{so then}$
    \begin{align*}\displaystyle
    A&=\frac{1}{2}\int_{\pi}^{2\pi}\left[(1)^2
    +(8(1+\sin{\theta}))^2\right] \, d\theta \\
    &=\frac{1}{2}\int_{\pi}^{2\pi}
    \left[
    1+64\theta + 128\sin{\theta}+64\sin^2{\theta}
    \right]d\theta \\
    &= \left|
    \theta +32\theta^2-128\cos{\theta}+32\theta-32\sin\theta\cos{\theta}
    \right|_{\pi}^{2\pi} \\
    &=16(5\pi-8)
    \end{align*}

    ok I am making errors all over the places but can't seem to straighten it out
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  2. #2
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    Re: Find the area of the specified region

    I am not sure why you had to consider $\frac{1}{2}r^2 d\theta$.
    In polar coordinates, area should be $\iint_{A} r drd\theta$.
    Therefore, you could proceed as attached image.
    Find the area of the specified region-test.jpg
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