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

Answer: 64

$A=\underset{\pi /4}{\overset{5\pi /4}{\int }}(\sin x-\cos x)dx$
$={(-\cos x-\sin x)|}_{\pi /4}^{5\pi /4}$
$=\left(-\left(\frac{-1}{\sqrt{2}}\right)-\left(\frac{-1}{\sqrt{2}}\right)\right)-\left(-\left(\frac{1}{\sqrt{2}}\right)-\left(\frac{1}{\sqrt{2}}\right)\right)$
$\Rightarrow A=\frac{2}{\sqrt{2}}+\frac{2}{\sqrt{2}}=2\sqrt{2}$
$\Rightarrow A^4=(2\sqrt{2}{)}^4=16\times 4=64$