Answer
$A_T=2\sqrt2$
Work Step by Step
$A_T = A_1 + A_2 $
$A_1=\int_0^{\frac{\pi}{4}}(cos(x)-sin(x))dx$
$A_1=[sin(x)+cos(x)]_0^{\frac{\pi}{4}}$
$A_1=[\frac{1}{\sqrt2}+\frac{1}{\sqrt2}]-[0+1]$
$A_1=\sqrt 2-1$
$A_2=\int_{\frac{\pi}{4}}^{\pi}(sin(x)-cos(x))dx$
$A_2=[-cos(x)-sin(x)]_{\frac{\pi}{4}}^{\pi}$
$A_2=[-(-1)-0]-[-\frac{1}{\sqrt2}-\frac{1}{\sqrt2}]$
$A_2=1+\sqrt 2$
$A_T=(\sqrt2-1)+(1+\sqrt2)=2\sqrt2$