### Quiz 5.6:

Question:

Find the area of one of the regions enclosed by $$\sin x$$ and $$\cos x$$

Solution:

There are infinitely many regions, but they are all congruent.
We need to first find two consecutive points of intersection, so we can get our limits of integration.
$$\sin x = \cos x$$
$$\tan x = 1$$
$$x= -\frac{3\pi}{4} , \frac{\pi}{4}$$

On this interval $$\cos x$$ is the larger function, so we have,
$$\int \limits _{-\frac{3\pi}{4}} ^{\frac{\pi}{4}} \left ( \cos x - \sin x \right ) dx$$
$$= \left [ \sin x + \cos x \right ] _{-\frac{3\pi}{4}} ^ {\frac{\pi}{4}}$$
$$= \left ( \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \right ) - \left (- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \right )$$
$$= 2\sqrt{2}$$