The area of the region $A = \{(x,y): 0 ≤ y ≤ x|x|+1\, and\, -1 ≤x≤1\}$ in square units, is

2

$y=x|x|+1,-1≤x≤1$

$⇒y=\left\{\begin{matrix}-x2+1,& -1≤x≤1\\x2+1,&0≤x≤1\end{matrix}\right.$

The required area A is given by

$A=\int\limits_{-1}^0(-x^2+1)dx+\int\limits_0^1(x^2+1)dx$

$⇒A=\left[-\frac{x^3}{3}+x\right]_{-1}^0+\left[\frac{x^3}{3}+x\right]_0^1=2$ sq. units.