Answer
The area of one petal:
$area = \frac{\pi }{{16}}$
Work Step by Step
In Exercise 9 of Section 12.4, we see that the "rose" traces out a complete curve for the interval $0 \le \theta \le 2\pi $. From the figure attached we see that it traces out two petals for the interval $0 \le \theta \le \frac{\pi }{2}$. So, by symmetry it traces out one petal for the interval $0 \le \theta \le \frac{\pi }{4}$.
Using Eq. (2) of Theorem 1 of Section 12.4, the area of one petal is
$area = \frac{1}{2}\cdot\mathop \smallint \limits_0^{\pi /4} {\sin ^2}\left( {4\theta } \right){\rm{d}}\theta $
Since ${\sin ^2}\left( {4\theta } \right) = \frac{1}{2}\left( {1 - \cos \left( {8\theta } \right)} \right)$, the integral becomes
$area = \frac{1}{4}\cdot\mathop \smallint \limits_0^{\pi /4} \left( {1 - \cos \left( {8\theta } \right)} \right){\rm{d}}\theta $
$area = \frac{1}{4}\left( {\theta - \frac{1}{8}\sin \left( {8\theta } \right)} \right)|_0^{\pi /4}$
$area = \frac{1}{4}\left( {\frac{\pi }{4}} \right) = \frac{\pi }{{16}}$
