Ola, boa tarde
Acredito que é \(S = 6\displaystyle\int_{0}^{\dfrac{\pi}{6}}{\dfrac{r^2}{2}d\theta= 3\displaystyle\int_{0}^{\dfrac{\pi}{6}}\sin^2{(3\theta)}d\theta}\)
Tem que desenvolver a integral a fazer subsituçao por ejemplo \(3\theta = \alpha\)
\(S= \displaystyle\int{\sin^2{\alpha}d\alpha}= \dfrac{1}{2}\displaystyle\int{(1- \cos{2\alpha})d\alpha}\)
\(S = \dfrac{1}{2}\left[\alpha - \dfrac{\sin{2\alpha}}{2}\right]= \dfrac{1}{2}\left[3\theta - \dfrac{\sin{6\theta}}{2}\right]_{o}^{\dfrac{\pi}{6}}= \dfrac{1}{2}\cdot\dfrac{3\pi}{6}= \dfrac{\pi}{4}\)
E coreto pues
https://pt.wikipedia.org/wiki/Rosa_polar
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