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

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

Application of Integrals

Solution:

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Intersection of curves $y=\sin x$ and $y=\cos x$ are $\frac{\pi}{4}, \frac{5 \pi}{4} \cdots$
Since, $\sin x \geq \cos x$ on $\left[\frac{\pi}{4}, \frac{5 \pi}{4}\right]$.
$\therefore A =\int\limits_{\pi / 4}^{5 \pi / 4}(\sin x-\cos x) d x$
$ =-[\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