The area bounded by the curves  $y=x$ and $y=x^3$ is 

The given curves are  $y=x$ and $y=x^3$ 

These two curves intersect at origin and $\left(1,1\right),\left(-1,-1\right)$

The area between the above two curves is as shown below

The area of the shaded region is the region bounded between the curves $y=x$ and  $y=x^3$ and the lines the curves $x=-1,x=1,y=0$

Hence, 

   $\begin{array}{rcl}A& =& {\int }_{-1}^1\left(x-x^3\right)dx\\ & =& {\left[\frac{x^2}{2}-\frac{x^4}{4}\right]}_{-1}^1\\ & =& \frac{1}{2}-\frac{1}{4}+\frac{1}{2}-\frac{1}{4}\\ & =& \frac{1}{2}\end{array}$

Therefore,  the area of the shaded region is $\boxed{0.5}$