Answer
$\dfrac{\pi\sqrt 2}{2}$
Work Step by Step
The area of the rectangle curve is=$(\sqrt 2)(\pi/4]=\dfrac{\pi\sqrt 2}{4}$
Need to find the area of the rectangle curve.
$A_1=-\int_{-\pi/4}^{0} \sec \theta \tan \theta d \theta$
This implies that
$[-\sec \theta]_{-\pi/4}^{0}=\sqrt 2-1$
Thus, the area of shaded region is: $\dfrac{\pi\sqrt 2}{4}+(\sqrt 2-1)$
Now, $A_2=\int_{\pi/4}^{0} \sec \theta \tan \theta d \theta$
This implies that
$[\sec \theta]_{\pi/4}^{0}=\sqrt 2-1$
Thus, the area of shaded region is: $A_2=\dfrac{\pi\sqrt 2}{4}-(\sqrt 2-1)$
Total area:$\int_{\pi/4}^{0} \sec \theta \tan \theta d \theta+ \dfrac{\pi\sqrt 2}{4}-(\sqrt 2-1)=\dfrac{\pi\sqrt 2}{2}$