Question

Choose the correct answer
The area bounded by the Y-axis, y = cos x and y = sin x, $0\le x\le \frac{\pi }{2}$ is
(a) $2(\sqrt{2}-1)$ (b) $\sqrt{2}-1$
(c) $\sqrt{2}+1$ (d) $\sqrt{2}$

Answer 1

The two curves y = sin x and y = cos x meet where sin x = cos x, i.e., where tan x = 1
$\Rightarrow x=\frac{\pi }{4}$
$\therefore$ Required area (shown in shaded region)
${\int }_0^{\frac{\pi }{4}}(cosx-sinx)dx=[sinx+cosx{]}_0^{\frac{\pi }{4}}=sin\frac{\pi }{4}+cos\frac{\pi }{4}-(0+1)=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-1$
$=\frac{2}{\sqrt{2}}-1=\sqrt{2}-1.$ So, the correct option is (b).