# Thread: Find the area of the specified region

1. ## 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

2. ## 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.