Chapter 10 - Parametric Equations and Polar Coordinates - 10.4 Areas and Lengths in Polar Coordinates - 10.4 Exercises - Page 713: 27

$$A =\pi$$

$$A=\frac{1}{2}\int_{0}^{\pi/3}(3cos\theta)^{2}d \theta-\frac{1}{2}\int_{0}^{\pi/3}(1+cos\theta)^{2}d \theta=\frac{1}{2}[3\theta+2sin2\theta-2sin\theta]_{0}^{\pi/3}$$ Or, $$A=\frac{\pi}{2}$$ Since, the area of the whole crescent is calculated as $$2 \cdot(\frac{\pi}{2})$$ Hence, $$A =\pi$$