Answer
$area = \frac{{25}}{4}\pi $.
Work Step by Step
We draw a circle of radius $5$. Then draw two rays that make the angles $\theta = \frac{\pi }{2}$ and $\theta=\pi$, respectively, with the $x$-axis. So, we obtain the region bounded by the circle and the rays.
Using Eq. (2) of Theorem 1, the area is
$area = \frac{1}{2}\cdot\mathop \smallint \limits_{\pi /2}^\pi {5^2}{\rm{d}}\theta = \frac{{25}}{2}\theta |_{\pi /2}^\pi = \frac{{25}}{4}\pi $.
