Chapter 10 - Section 10.5 - Areas and Lengths in Polar Coordinates - Exercises - Page 584: 1

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

The area of a shaded region can be found as: $A=\dfrac{1}{2}\int_p^q r^2 d \theta$ Here, we have $A=\dfrac{1}{2}\int_{0}^{\pi} (\theta)^2 d \theta$ $\dfrac{1}{2}[\dfrac{\theta^3}{3}]_{0}^{\pi}=(\dfrac{1}{6})[\pi^3-0]$ Thus, $A=\dfrac{\pi^3}{6}$