What is the area bounded by the curves $|y|=1-x^2$?

The equation $|y|=1-x^2$ implies:$y=1-x^2$ for $y\ge 0$$-y=1-x^2⟹y=x^2-1$ for $y<0$ These represent two parabolas opening downwards and upwards, intersecting at $x=\pm 1$.The total area is symmetric about both axes.
$$\text{Area}=2{\int }_{-1}^1(1-x^2)dx=4{\int }_0^1(1-x^2)dx$$
$$\text{Area}=4[x-\frac{x^3}{3}{]}_0^1=4(1-\frac{1}{3})=4(\frac{2}{3})=\frac{8}{3}$$