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 $\mathrm{A}^{4}$ is equal to______.
Solution:
$A=\int_{\pi / 4}^{5 \pi / 4}(\sin x-\cos x) d x$
$=\left.(-\cos x-\sin x)\right|_{\sqrt / 4} ^{5 \pi / 4}$
$=\left(-\left(\frac{-1}{\sqrt{2}}\right)-\left(\frac{-1}{\sqrt{2}}\right)\right)-\left(-\left(\frac{1}{\sqrt{2}}\right)-\left(\frac{1}{\sqrt{2}}\right)\right)$
$\Rightarrow A=\frac{2}{\sqrt{2}}+\frac{2}{\sqrt{2}}=2 \sqrt{2}$
$\Rightarrow \mathrm{A}^{4}=(2 \sqrt{2})^{4}=16 \times 4=64$
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