$$A = \int\limits_{{\pi \over 4}}^{{{5\pi } \over 4}} {(\sin x - \cos x)dx} $$

$$= [ - \cos x - \sin x]_{\pi /4}^{5\pi /4}$$

$$ = - \left[ {\left( {\cos {{5\pi } \over 4} + \sin {5\pi \over 4}} \right) - \left( {\cos {\pi \over 4} + \sin {\pi \over 4}} \right)} \right]$$

$$ = - \left[ {\left( { - {1 \over {\sqrt 2 }} - {1 \over {\sqrt 2 }}} \right) - \left( {{1 \over {\sqrt 2 }} + {1 \over {\sqrt 2 }}} \right)} \right]$$

$$ = {4 \over {\sqrt 2 }} = 2\sqrt 2 $$

$$ \Rightarrow {A^4} = {\left( {2\sqrt 2 } \right)^4} = 64$$