Mathematics - Area Under Curves Question with Solution | TestHub

$A={\int }_0^{\pi /2}\left(\right(\cos x+\sin x)-|\cos x-\sin x\left|\right)dx$

$A={\int }_0^{\pi /4}\left(\right(\cos x+\sin x)-(\cos x-\sin x\left)\right)dx+{\int }_{\pi /4}^{\pi /2}\left(\right(\cos x+\sin x)-(\sin x-\cos x\left)\right)dx$

$A=2{\int }_0^{\pi /4}\sin xdx+2{\int }_{\pi /4}^{\pi /2}\cos xdx$

$=2\left(-\frac{1}{\sqrt{2}}+1\right)+2\left(1-\frac{1}{\sqrt{2}}\right)=4-2\sqrt{2}=2\sqrt{2}(\sqrt{2}-1)$