Mathematics - Area Under Curves Question with Solution | TestHub

The given curves are $y=\sin x+\cos x \& y=\left|\cos x-\sin x\right|$

${\text{y}}_1=\text{sin x}+\text{cos x}=\sqrt{2}\text{ sin}\left(\text{x}+\frac{\pi }{4}\right)$
${\text{y}}_2=\sqrt{2}\left|\text{sin}\left(\frac{\pi }{4}-\text{x}\right)\right|$
So, Required Area$=\underset{0}{\overset{\frac{\pi }{4}}{\int }}\left(\left(\text{sin x}+\text{cos x}\right)-\left(\text{cos x}-\text{sin x}\right)\right)\text{dx}+\underset{\frac{\pi }{4}}{\overset{\frac{\pi }{2}}{\int }}\left(\left(\text{sin x}+\text{cos x}\right)-\left(\text{sin x}-\text{cos x}\right)\right)\text{dx}$
$2{\int }_0^{\frac{\mathrm{\pi }}{4}}\sin xdx+2{\int }_{\frac{\mathrm{\pi }}{4}}^{\frac{\mathrm{\pi }}{2}}\cos xdx$
$=4-2\sqrt{2}=2\sqrt{2}\left(\sqrt{2}-1\right)$.