Find the area of the region specified in polar coordinates.
-one petal of the rose curve $r = 9 \cos 3 \theta$
A) $\frac { 81 } { 2 } \pi$
B) $\frac { 81 } { 4 } \pi$
C) $\frac { 27 } { 4 } \pi$
D) $\frac { 27 } { 2 } \pi$

Question

Quizplus · Accepted Answer

One petal of the rose curve is from $	heta=0$ to $	heta=\frac{\pi}{6}$. To find the area, we integrate $\frac{1}{2}r^2 d	heta$ from $	heta=0$ to $	heta=\frac{\pi}{6}$. \begin{align*}
\int_0^{\frac{\pi}{6}}\frac{1}{2}(9\cos3	heta)^2 d	heta&=\frac{81}{4}\int_0^{\frac{\pi}{6}}\frac{1+\cos6	heta}{2} d	heta\
&=\frac{81}{4}\left(\frac{	heta}{2}+\frac{1}{12}\sin6	hetaight)\bigg|_0^{\frac{\pi}{6}}\
&=\frac{27}{4}\left(\pi+2\sin\frac{\pi}{2}-\sqrt3ight)\
&=\frac{27}{4}\left(\pi+2-\sqrt3ight)
\end{align*} Therefore, the answer is $\boxed{	extbf{(C)}\ \frac{27}{4}\pi}$.

Find the Area of the Region Specified in Polar Coordinates r=9cos⁡3θr = 9 \cos 3 \thetar=9cos3θ

Find the Area of the Region Specified in Polar Coordinates $r = 9 \cos 3 \theta$