The area enclosed by the curves y = x + x and y = |cos x - sin x|...

The given curves are $y = \sin x + \cos x \text{ and } y = \left|\cos x - \sin x\right|$

{\text{y}}_{1} = \text{sin x} + \text{cos x} = \sqrt{2} \text\sin \left(\text{x} + \dfrac{\pi}{4}\right)
{\text{y}}_{2} = \sqrt{2} | \text\sin \left(\dfrac{\pi}{4} - \text{x}\right) |
So, Required Area$= \mathop{∫}_{0}^{\dfrac{\pi}{4}}\left(\left(\text{sin x} + \text{cos x}\right) - \left(\text{cos x} - \text{sin x}\right)\right)\text{dx} + \mathop{∫}_{\dfrac{\pi}{4}}^{\dfrac{\pi}{2}}\left(\left(\text{sin x} + \text{cos x}\right) - \left(\text{sin x} - \text{cos x}\right)\right)\text{dx}$
$2{∫}_{0}^{\dfrac{\pi}{4}}sinxdx + 2{∫}_{\dfrac{\pi}{4}}^{\dfrac{\pi}{2}}cosxdx$
$= 4 - 2\sqrt{2} = 2\sqrt{2}\left(\sqrt{2} - 1\right)$.