Question

The graphs 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 1

$A=\underset{\frac{\pi }{4}}{\overset{\frac{5\pi }{4}}{\int }}(\sin x-\cos x)dx=[-\cos x-\sin x{]}_{\pi /4}^{5\pi /4}=-[(\cos \frac{5\pi }{4}+\sin \frac{5\pi }{4})-(\cos \frac{\pi }{4}+\sin \frac{\pi }{4})]$

$=-[(-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}})-(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}})]$
$=\frac{4}{\sqrt{2}}=2\sqrt{2}$
$\Rightarrow A^4=(2\sqrt{2}{)}^4=64$