Answer
$A=\frac{3}{4\pi}$
Work Step by Step
Setting up the integral using the formula for the area of a region in polar coordinates, we get:
$$\frac{1}{2}\int_{\pi/2}^{2\pi}\bigg(\frac{1}{\theta}\bigg)^2 d\theta$$
We can rewrite this expression as:
$$\frac{1}{2}\int_{\pi/2}^{2\pi}\theta^{-2} d\theta$$
Integrating, we get:
$$\frac{1}{2}(-\theta^{-1})\bigg\rvert_{\pi/2}^{2\pi}=\frac{1}{2}\Bigg(-\frac{1}{2\pi}-\bigg(-\frac{1}{\pi/2}\bigg)\Bigg)=\frac{3}{4\pi}$$
