question_answer

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

A)  6                                            

B)  \[\frac{1}{3}\]

C)  \[\frac{2}{3}\]                                  

D)  None of these

Correct Answer: C

Solution :

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