question_answer

The area of the region bounded by the curve \[y=x\left| x \right|\], x-axis and the ordinates \[\operatorname{x} = 1, x = -1\] is given by:

A) zero                  

B)        \[\frac{1}{3}\]

C) \[\frac{2}{3}\]              

D)        1

Correct Answer: C

Solution :

The area of the region bounded by the curve \[y=f(x)\] and the ordinates \[\operatorname{x} = a, \,x = b\] is given by \[Area=\left| \int{_{a}^{b}\,y\,dx} \right|\] According to the question, \[y=x\left| x \right|=\left\{ \begin{matrix}    {{x}^{2}},\,\,x\ge 0  \\    -\,{{x}^{2}},\,\,x<0  \\ \end{matrix} \right.\] Required area \[= area of region OAB + area of region OCD\] \[= 2 \times  Area of region OAB\] \[=\,\,\,2\int_{0}^{1}{{{x}^{2}}dx=\frac{2}{3}}\,sq.units\]