The area bounded by the y-axis, \( y=\cos x \) and \( y=\sin x \) , when \( 0\le x\le\displaystyle\frac{\pi}{4} \) , is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \sqrt{2}-1 \)
The curves meet where \( \cos x=\sin x \), at \( x=\frac{\pi}{4} \). For \( 0\le x\le\frac{\pi}{4} \), cos x ≥ sin x, so y = cos x is the upper curve and the region runs from the y-axis (x = 0) to x = π/4.
\( \displaystyle \text{Area}=\int_{0}^{\pi/4}(\cos x-\sin x)\,dx \)
\( \displaystyle =\Big[\sin x+\cos x\Big]_{0}^{\pi/4} \)
\( \displaystyle =\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}\right)-(0+1) \)
\( \displaystyle =\sqrt{2}-1 \)
NCERT Exemplar:
Application of Integrals
