Question

The area of the region bounded by the $Y$-axis, $y=\cos x$ and $y=\sin x,0\le x\le \frac{\pi }{2}$ is

• A. $\sqrt{2}$ sq units
• B. $(\sqrt{2}+1)$ sq units
• C. $(\sqrt{2}-1)$ sq units
• D. $(2\sqrt{2}-1)$ sq units

Answer 1

Answer: (C)

Let us first draw the graph of $\cos x$ and $\sin x$ between 0 to $\frac{\pi }{2}$ and shade the region bounded by them with $Y$-axis.

$\therefore$ Area of shaded region $=$ Area $(OACBO-$ Area $(OCBO)$
$=\underset{0}{\overset{\pi }{\int }}(\cos x-\sin x)dx$
$=[\sin x+\cos x{]}_0^{\pi /4}$
$=[\left(\sin \frac{\pi }{4}+\cos \frac{\pi }{4}\right)-\{(\sin (0)+\cos (0)\}]$
$=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-1=(\sqrt{2}-1)$ sq units