Answer
$2 \sqrt 2$
Work Step by Step
Let us consider that two continuous functions $f(x)$ and $g(x)$ with $f(x)\geq g(x)$ on the interval $[a,b]$ . The area (A) of the region bounded by the graph of $f(x)$ and $g(x)$ on the interval $[a,b]$ can be calculated as: $Area(A)=\int_a^b [f(x)-g(x)] \ dx$
The given curves are $y=\cos x$ and $y=\sin x$.The intersection points of these curves satisfy the given equations are: $\cos x=\sin x \implies x=\dfrac{\pi}{4} \ \text{and}\ x=\dfrac{5 \pi}{4}$ . These points will be the lower and upper limit of the integration.
Thus, the area of the region is:
$A=\int_a^b [f(x)-g(x)] \ dx= \int_{\dfrac{\pi}{4}}^{\dfrac{5\pi}{4}} [\sin x-\cos x] \ dx\\= [-\cos x-\sin x]_{\dfrac{\pi}{4}}^{\dfrac{5\pi}{4}} \\ =-\cos \dfrac{5 \pi}{4} -\sin \dfrac{5 \pi}{4} +\cos \dfrac{ \pi}{4} +\sin \dfrac{ \pi}{4} \\=\dfrac{1}{\sqrt 2}+\dfrac{1}{\sqrt 2}+\dfrac{1}{\sqrt 2}+\dfrac{1}{\sqrt 2} \\=2 \sqrt 2$
