Find the area of the region specified in polar coordinates.
-one petal of the rose curve $r = 10 \sin 2 \theta$
A) $25 \pi$
B) $50 \pi$
C) $\frac { 50 } { 3 } \pi$
D) $\frac { 25 } { 2 } \pi$

Question

Quizplus · Accepted Answer

One petal of the rose curve is traced out as $	heta$ goes from $0$ to $\pi/4$, so we want the area of the region enclosed by one petal, the x-axis, and the vertical line $	heta=\pi/4$, which is given by the integral \begin{align*}
\frac12\int_0^{\pi/4}\left(\frac{10}{2}\sin 2	hetaight)^2\,d	heta &= \frac{100}{4}\int_0^{\pi/4}\sin^2 2	heta\,d	heta \
&= \frac{25}{2}\int_0^{\pi/4}\frac{1-\cos 4	heta}{2}\,d	heta \quad	ext{(use trig identity)}\
&= \frac{25}{2}\cdot\frac{1}{2}\left(	heta-\frac14\sin 4	hetaight)\bigg|_0^{\pi/4} \
&= \frac{25}{2}\cdot\frac{1}{2}\left(\frac{\pi}{4}-\frac{1}{4}\sin\frac{\pi}{2}ight) \
&= \boxed{\frac{25}{2}\pi}.
\end{align*}

Find the Area of the Region Specified in Polar Coordinates r=10sin⁡2θr = 10 \sin 2 \thetar=10sin2θ

Find the Area of the Region Specified in Polar Coordinates $r = 10 \sin 2 \theta$