Area of region defined by 

$A_1=\left\{\begin{array}{r}(x,y):x\ge 0,y\ge 0,2x+2y-x^2\\ -y^2>1>x+y\end{array}\right\}$ is $\left|A_1\right|$.

Similarly, $\left|A_2\right|$ is the area of region defined by 

$A_2=\left\{\left(x,y\right):x\ge 0,y\ge 0,x+y>1>x^2+y^2\right\}$

Here, $\left|A_1\right|=\left|A_2\right|$

and area $\left|A_3\right|$ is the area of region defined by $A_3=\left\{\left(x,y\right):x\ge 0,y\ge 0,x+y>1>x^3+y^3\right\}$

then $\left|A_3\right|>\left|A_2\right|=\left|A_1\right|$