question_answer

The area bounded by the parabola \[y=4{{x}^{2}},y=\frac{{{x}^{2}}}{9}\]and the line \[y=2\] is

A) \[\frac{5\sqrt{2}}{3}\text{sq}\text{.}\,\text{units}\]         

B) \[\frac{10\sqrt{2}}{3}\text{sq}\text{.}\,\text{units}\]

C) \[\frac{15\sqrt{2}}{3}\text{sq}\text{.}\,\text{units}\]

D) \[\frac{20\sqrt{2}}{3}\text{sq}\text{.}\,\text{units}\]

Correct Answer: D

Solution :

[d]: Given, \[y=4{{x}^{2}}\]\[\Rightarrow {{x}^{2}}=\frac{y}{4}\]                             ...(i) \[{{x}^{2}}=9y\]  ...(ii) and \[y=2\]                                            ...(iii) \[\therefore \]Area bounded by the above three curves \[=2\int\limits_{0}^{2}{\left( 3\sqrt{y}-\frac{\sqrt{y}}{2} \right)dy=2\int\limits_{0}^{2}{\frac{5}{2}}\sqrt{y}dy=2\times \frac{5}{2}\left[ \frac{2{{y}^{3/2}}}{3} \right]_{0}^{2}}\]\[=\frac{10}{3}[2\sqrt{2}-0]=\frac{20\sqrt{2}}{3}\]sq. unit