Chapter 11: Parametric Equations and Polar Coordinates - Section 11.5 - Areas and Lengths in Polar Coordinates - Exercises 11.5 - Page 670: 1

$A=\dfrac{\pi^3}{6}$

The area of a shaded region is given as: $A=\dfrac{1}{2}\int_p^q r^2 d \theta$ consider $A=\dfrac{1}{2}\int_{0}^{\pi} (\theta)^2 d \theta$ $\implies A= (\dfrac{1}{2}) [\dfrac{\theta^3}{3}]_{0}^{\pi}=(\dfrac{1}{6})[\pi^3-0]$ Thus, $A=\dfrac{\pi^3}{6}$