question_answer

Area between the curve \[y = 2{{x}^{4}}-{{x}^{2}},x\] axis and the ordinates of two minima of curve is

A) \[\frac{7}{120}\]

B) \[\frac{9}{120}\]

C)  \[\frac{11}{120}\]

D) \[\frac{13}{120}\]

Correct Answer: A

Solution :

The graph of \[y=2{{x}^{4}}-{{x}^{2}}\] is shown below:

Minimum occurs at \[x=\pm \frac{1}{2}\]

\[\therefore \]Desired area

\[=2\int\limits_{0}^{1/2}{\left( -y \right)dx=2\int\limits_{0}^{1/2}{\left( {{x}^{2}}-2{{x}^{4}} \right)}dx}\]

\[=2\left[ \frac{{{x}^{3}}}{3}-\frac{2{{x}^{5}}}{5} \right]_{0}^{1/2}=2\left[ \frac{1}{24}-\frac{1}{80} \right]=\frac{7}{120}\]