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

Enclosed area between the curves:

Given : 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 $A$.

Let's draw the graph :

$A={\int }_{\frac{\mathrm{\pi }}{4}}^{\frac{5\mathrm{\pi }}{4}}\left(\sin x-\cos x\right)dx$

$\Rightarrow$$A={\left[-\cos x-\sin x\right]}_{\frac{\mathrm{\pi }}{4}}^{\frac{5\mathrm{\pi }}{4}}$

$\Rightarrow$$A=-\left[\left(\cos \frac{5\mathrm{\pi }}{4}+\sin \frac{5\mathrm{\pi }}{4}\right)-\left(\cos \frac{\mathrm{\pi }}{4}+\sin \frac{\mathrm{\pi }}{4}\right)\right]$

$\Rightarrow$$A=-\left[\left(-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}\right)-\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}\right)\right]$

$\Rightarrow$$A=\frac{4}{\sqrt{2}}$

$\Rightarrow$$A=2\sqrt{2}$

$\Rightarrow$$A^4={\left(2\sqrt{2}\right)}^4$

$\Rightarrow$$A^4=64$

Hence $A^4$ is equal to $64$.