Chapter 11: Parametric Equations and Polar Coordinates - Section 11.5 - Areas and Lengths in Polar Coordinates - Exercises 11.5 - Page 670: 3

$18 \pi$

Consider $A=\dfrac{1}{2}\int_m^n (r^2) d \theta=\dfrac{1}{2}(2)\int_{0}^{\pi} (4+2 \cos \theta)^2 d \theta$ $\implies A= \int_{0}^{\pi} 4\cos^2 \theta d \theta+\int_{0}^{\pi} 16 \cos \theta d \theta+\int_{0}^{\pi} 16 \cos \theta d \theta$ $\implies A=(\dfrac{1}{2}) \int_{0}^{\pi} 4(1+\cos2 \theta) d \theta+(16) \int_{0}^{\pi} \cos \theta d \theta+(16) \int_{0}^{\pi} \cos \theta d \theta=18 \pi$