Question

The sine and cosine curves intersect infinitely many times giving bounded regions of equal areas. The area of one of such region is

• A. $\sqrt{2}$ sq units
• B. $2\sqrt{2}$ sq units
• C. $3\sqrt{2}$ sq units
• D. $4\sqrt{2}$ sq units

Answer 1

Answer: (B)

Intersection of curves $y=\sin x$ and $y=\cos x$ are $\frac{\pi }{4},\frac{5\pi }{4}\cdots$
Since, $\sin x\ge \cos x$ on $\left[\frac{\pi }{4},\frac{5\pi }{4}\right]$.
$\therefore A=\underset{\pi /4}{\overset{5\pi /4}{\int }}(\sin x-\cos x)dx$
$=-[\cos x+\sin x{]}_{\pi /4}^{5\pi /4}$
$=-\left[\left(\cos \frac{5\pi }{4}+\sin \frac{5\pi }{4}\right)-\left(\cos \frac{\pi }{4}+\sin \frac{\pi }{4}\right)\right]$
$=-\left[\left(\frac{-1}{2}-\frac{1}{\sqrt{2}}\right)-\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}\right)\right]=-[-\sqrt{2}-\sqrt{2}]$
$=2\sqrt{2}$ sq units