Question

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

• A. $0$
• B. $\frac{1}{3}$
• C. $\frac{2}{3}$
• D. $\frac{4}{3}$

Answer 1

Answer: (C)

$\frac{2}{3}$

$y=x^2$ if $x>0$ and $y=-x^2$ if $x<0$
Required area $={\int }_{-1}^{1}ydx$

$={\int }_{-1}^{1}x|x|dx$

$={\int }_{-1}^0{x}^{2}dx+{\int }_0^1{x}^{2}dx$

$={\left[\frac{x^3}{3}\right]}_{-1}^0+{\left[\frac{x^3}{3}\right]}_0^1$

$=-(-\frac{1}{3})+\frac{1}{3}$

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

Thus, the correct answer is C.