Question

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

• A. zero
• B. $\frac{1}{3}$
• C. $\frac{2}{3}$
• D. 1

Answer 1

Answer: (C)

$\frac{2}{3}$

Answer 2

The area of the region bounded by the curve $y = f (x)$ and the ordinates $x = a, x = b$ is given by
Area $= \left|\int\limits^{b}_{a} y\, dx\right|$
According to the question,
$y =x\left|x\right| = \begin{cases} x^2 &, x \ge 0 \\\ -x^2 & x < 0 \end{cases}$

= area of region $OAB$ + area of region $OCD$
$= 2 \times$ Area of region $OAB$
$= 2 \int\limits^{1}_{0} x^2 dx = \frac{2}{3}$ s units.