Area bounded by the region $R\equiv \{(x,y):y^2\le x\le |y|\}$ is:

The region is defined by $y^2\le x$ (interior of parabola $x=y^2$) and $x\le |y|$ (region between $y$ -axis and lines $x=y,x=-y$).The curves intersect where $y^2=|y|$.For $y>0$: $y^2=y⟹y=1,x=1$.For $y<0$: $y^2=-y⟹y=-1,x=1$.The region is symmetric about the $x$ -axis.Area $=2{\int }_0^1(\text{right curve}-\text{left curve})dy$.Wait, the inequality is $y^2\le x\le |y|$.This means $x$ is between the parabola $x=y^2$ and the line $x=|y|$.Actually, for $0<y<1$, $y^2<y$, so $x$ is between $y^2$ and $y$.Area $=2{\int }_0^1(y-y^2)dy=2[y^2/2-y^3/3{]}_0^1$ Area $=2[1/2-1/3]=2[1/6]=1/3$.