Answer
$$\sqrt 2 $$
Work Step by Step
$$\eqalign{
& \int_{ - \pi /4}^{\pi /4} {\int_{\sin x}^{\cos x} {dy} dx} \cr
& {\text{Integrate with respect to }}y \cr
& = \int_{ - \pi /4}^{\pi /4} {\left[ y \right]_{\sin x}^{\cos x}dx} \cr
& = \int_{ - \pi /4}^{\pi /4} {\left( {\cos x - \sin x} \right)dx} \cr
& {\text{Integrate with respect to }}x \cr
& = \left[ {\sin x + \cos x} \right]_{ - \pi /4}^{\pi /4} \cr
& {\text{Evaluating}} \cr
& = \left[ {\sin \left( {\frac{\pi }{4}} \right) + \cos \left( {\frac{\pi }{4}} \right)} \right] - \left[ {\sin \left( { - \frac{\pi }{4}} \right) + \cos \left( { - \frac{\pi }{4}} \right)} \right] \cr
& = \left( {\frac{{\sqrt 2 }}{2} + \frac{{\sqrt 2 }}{2}} \right) - \left( { - \frac{{\sqrt 2 }}{2} + \frac{{\sqrt 2 }}{2}} \right) \cr
& = \sqrt 2 \cr} $$
