Question

The area bounded by the two curves $y=\sin x,y=\cos x$ and the $X$-axis in the first quadrant $[0,\frac{\pi }{2}]$ is

• A. $2-\sqrt{2}$ sq. units
• B. $2+\sqrt{2}$ sq,. units
• C. $2(\sqrt{2}-1)$ sq. units
• D. $4$ sq. units

Answer 1

Answer: (A)

$2-\sqrt{2}$ sq. units

The area bounded by the curves $y=\sin x,y=\cos x$ and $X$-axis in $[0,\frac{\pi }{2}]$ is given by
$A=\stackrel{\pi /4}{\underset{0}{\int }}\sin xdx+\stackrel{\pi /2}{\underset{\pi /4}{\int }}\cos xdx$

$=-[\cos x{]}_0^{\pi /4}+[\sin x{]}_{\pi /4}^{\pi /2}$

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

$=(2-\sqrt{2})squnits$