Sketch the graph of $y = x|x|$ and hence find the area bounded by this curve, X-axis and the ordinates $x = -2$ and $x = 2$, using integration.

$\frac{16}{3}$ sq units

The correct answer is Option (3) → $\frac{16}{3}$ sq units

$y = x|x|$

$y = x^2$ if $x > 0$

and $y = -x^2$ if $x < 0$

$\text{Required Area} = \int\limits_{-2}^{2} y dx$

$= \int\limits_{-2}^{2} x|x| dx$

$= \left|\int\limits_{-2}^{0} -x^2 dx \right| + \int\limits_{0}^{2} x^2 dx$

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

$= \left[ 0 - \left[ \frac{(-2)^3}{3} \right] \right] + \left[ \frac{2^3}{3} - 0 \right]$

$= \frac{8}{3} + \frac{8}{3} = \frac{16}{3} \text{ units}$