Answer
$\dfrac{\pi\sqrt 2}{2}$
Work Step by Step
The area of the rectangle curve is given as: $A_1=-\int_{-\pi/4}^{0} \sec \theta \tan \theta d \theta=[-\sec \theta]_{-(\pi/4)}^{0}=\sqrt 2-1$
Now, the area of shaded region will be: $\dfrac{\pi\sqrt 2}{4}+(\sqrt 2-1)$
Next, we have $A_2=\int_{(\pi/4)}^{0} \sec \theta \tan \theta d \theta=[\sec \theta]_{(\pi/4)}^{0}=\sqrt 2-1$
Now, the area of shaded region is: $A_2=\dfrac{\pi\sqrt 2}{4}-(\sqrt 2-1)$
So, the total area is: $A=A_1+A_2$
or, $A=\dfrac{\pi\sqrt 2}{4}+(\sqrt 2-1)+ \dfrac{\pi\sqrt 2}{4}-(\sqrt 2-1)=\dfrac{\pi\sqrt 2}{2}$
