Solución

La gráfica de $r = 3sen (2\theta)$ es la siguiente.

Figura 1.31 Gráfica de $r = 3sen (2\theta)$ .

\begin{aligned} A &= \frac12\int_{\alpha}^{\beta}(f(\theta))^2d\theta\\ &= \frac12\int_{0}^{\pi/2}(3sen(2\theta))^2d\theta\\ &= \frac12\int_{0}^{\pi/2} 9sen^2(2\theta)d\theta \end{aligned}

Para evaluar esta integral, usa la fórmula $sen^2\alpha = (1 − cos (2\alpha))/2$ con $\alpha = 2\theta$ :

\begin{aligned} A &= \frac{1}{2}\int_{0}^{\pi/2} 9sen^2(2\theta)d\theta\\ &= \frac92\int_{0}^{\pi/2}\frac{1-cos(4\theta)}{2}d\theta\\ &= \frac94 \left(\int_{0}^{\pi/2} 1-cos(4\theta)d\theta\right)d\theta\\ &= \frac94 \left(\theta-\frac{sen(4\theta)}{4}\right)_{0}^{\pi/2}\\ &= \frac94\left(\frac{\pi}{2}-\frac{sen(2\pi)}{4}\right) - \frac94\left(0-\frac{sen 4(0)}{4}\right)\\ &= \frac{9\pi}{8} \end{aligned}