Answer
$$A = 2\sqrt 2 $$
Work Step by Step
$$\eqalign{
& y = \sin x,{\text{ }}y = \cos x,{\text{ }}\frac{\pi }{4} \leqslant x \leqslant \frac{{5\pi }}{4} \cr
& \sin x \geqslant \cos x{\text{ on the interval }}\left( {\frac{\pi }{4},\frac{{5\pi }}{4}} \right) \cr
& {\text{From the graph and using symmetry properties, we obtain }} \cr
& A = \int_{\pi /4}^{5\pi /4} {\left( {\sin x - \cos x} \right)} dx \cr
& {\text{Integrate and evaluate}} \cr
& A = \left[ { - \cos x - \sin x} \right]_{\pi /4}^{5\pi /4} \cr
& A = - \left[ {\cos \left( {\frac{{5\pi }}{4}} \right) + \sin \left( {\frac{{5\pi }}{4}} \right)} \right] + \left[ {\cos \left( {\frac{\pi }{4}} \right) + \sin \left( {\frac{\pi }{4}} \right)} \right] \cr
& A = - \left( { - \sqrt 2 } \right) + \left( {\sqrt 2 } \right) \cr
& A = 2\sqrt 2 \cr} $$
