The area bounded by curves $\mathrm{y}={\mathrm{x}}^3;\mathrm{y}={\left(\mathrm{x}\right)}^{1/3}$

$\mathrm{y}={\mathrm{x}}^3$ and $\mathrm{y}={\mathrm{x}}^{1/3}$ are mirror images of each other about the line $\mathrm{y}=\mathrm{x}$ as shown below:

Clearly the bounded area lies in 1^st and III quadrant and is equal to
$4\left[\underset{0}{\overset{1}{\int }}(\mathrm{x}-{\mathrm{x}}^3)\mathrm{dx}\right]=4{\left[\frac{{\displaystyle {\mathrm{x}}^2}}{{\displaystyle 2}}-\frac{{\displaystyle {\mathrm{x}}^4}}{{\displaystyle 4}}\right]}_0^1=4\left[\frac{{\displaystyle 1}}{{\displaystyle 2}}-\frac{{\displaystyle 1}}{{\displaystyle 4}}\right]=1$ square units.