Answer
$$A = 4\sqrt 2 $$
Work Step by Step
$$\eqalign{
& {\text{From the graph we can note that the area is given by}} \cr
& A = \int_0^{\pi /4} {\left( {\cos x - \sin x} \right)} dx + \int_{\pi /4}^{5\pi /4} {\left( {\sin x - \cos x} \right)} dx \cr
& {\text{ }} + \int_{5\pi /4}^{2\pi } {\left( {\cos x - \sin x} \right)} dx \cr
& {\text{Integrate}} \cr
& A = \left[ {\sin x + \cos x} \right]_0^{\pi /4} + \left[ { - \cos x - \sin x} \right]_{\pi /4}^{5\pi /4} + \left[ {\sin x + \cos x} \right]_{5\pi /4}^{2\pi } \cr
& {\text{Evaluating}} \cr
& A = \left( {\sin \frac{\pi }{4} + \cos \frac{\pi }{4}} \right) - \left( {\sin 0 + \cos 0} \right) + \left( { - \cos \frac{{5\pi }}{4} - \sin \frac{{5\pi }}{4}} \right) \cr
& {\text{ }} - \left( { - \cos \frac{\pi }{4} - \sin \frac{\pi }{4}} \right) + \left( {\sin 2\pi + \cos 2\pi } \right) - \left( {\sin \frac{{5\pi }}{4} + \cos \frac{{5\pi }}{4}} \right) \cr
& {\text{Simplifying}} \cr
& A = \frac{{\sqrt 2 }}{2} + \frac{{\sqrt 2 }}{2} - 1 + \frac{{\sqrt 2 }}{2} + \frac{{\sqrt 2 }}{2} + \frac{{\sqrt 2 }}{2} + \frac{{\sqrt 2 }}{2} + 1 + \frac{{\sqrt 2 }}{2} + \frac{{\sqrt 2 }}{2} \cr
& A = 8\left( {\frac{{\sqrt 2 }}{2}} \right) \cr
& A = 4\sqrt 2 \cr} $$
