The area bounded by the curve y = -x|x|, x-axis and the ordinates x = -1 and x = 1 is given by :

$\frac{2}{3}$

y = -x|x|

$y=\left\{\begin{array}{rl}-x^2 & x \geq 0 \\ x^2 & x<0\end{array}\right.$

plotting graph 

x-axis

x = -1

and x = +1

area of region I = area of region II (by symmetry)

So total area = 2 × area of region I

$=2 x \int\limits_{-1}^0 x^2 d x$

$2 \times\left[\frac{x^3}{3}\right]_{-1}^0$

$= \frac{2}{3}$ sq units