The graph of sine and cosine functions intersect each other at a number of points, and bet…

Question

The graph of sine and cosine functions intersect each other at a number of points, and between two consecutive points of intersection the two graphs enclose the same area $A$. Then $A^4$ is equal to \_\_\_\_\_\_\_\_.

Answer: 64

Accepted Answer

Between consecutive intersections $x=\pi/4$ and $x=5\pi/4$:

$$A = \int_{\pi/4}^{5\pi/4}(\sin x-\cos x)\,dx = [-\cos x-\sin x]_{\pi/4}^{5\pi/4} = 2\sqrt2$$

$$A^4 = (2\sqrt2)^4 = 64$$

Solutions · 1