question_answer

The area bounded by the curve y = x \[\left| x \right|\], x-axis and the ordinates x = 1, \[x=-1\] is given by

A) 0         

B) 1/3

C) 2/3                   

D) None of these

Correct Answer: C

Solution :

[c] Required area\[=\left| \int_{-1}^{0}{x|x|dx} \right|\] \[=\left| \left| \int_{-1}^{0}{x|x|dx} \right| \right|+\left| \int_{1}^{0}{x|x|dx}= \right|\left| \int_{-1}^{0}{{{x}^{2}}dx} \right|+\int_{0}^{1}{{{x}^{2}}dx}\]\[=\left| \left[ -\frac{{{x}^{3}}}{3} \right]_{-1}^{0} \right|-{{\left[ \frac{{{x}^{3}}}{3} \right]}^{1}}_{0}=\frac{1}{3}+\frac{1}{3}+\frac{2}{3}\]