The area of the region bounded by the curve $y=x\left|x\right|,$ x-axis and the ordinates $x=1,x=-1$ is given by:

The area of the region bounded by the curve $y=f\left(x\right)$ and the ordinates $x=a, x=b$ is given by $Area=\left|{\int }_a^{b}ydx\right|$ 
According to the question, $y=x\left|x\right|=\{\begin{array}{l}{x}^2,x\ge 0\\ -x^2,x<0\end{array}$  
 
Required area = area of region OAB + area of region OCD
$=2\times areaofregionOAB=2{\int }_0^1{x}^{2}dx=\frac{2}{3}sq.units$