Question

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

• A. $\frac{5\sqrt{2}}{3}$ sq units
• B. $\frac{10\sqrt{2}}{3}$ sq units
• C. $\frac{15\sqrt{2}}{3}$ sq units
• D. $\frac{20\sqrt{20}}{3}$ sq units

Answer 1

The correct option is D $\frac{20\sqrt{20}}{3}$ sq units

$y=4x^2\dots \left(i\right)$
$y=\frac{x^2}{4}\dots \left(ii\right)$
$\therefore A={\int }_0^2\left[\begin{array}{c}\frac{\sqrt{y}}{2}-3\sqrt{y}\end{array}\right]dy$
$=\left(\begin{array}{c}\frac{1}{2}-3\end{array}\right){\int }_0^2\sqrt{y}dy$
$=\left(\begin{array}{c}\frac{-5}{2}\end{array}\right).\frac{2}{3}{\left[\begin{array}{c}{y}^{3/2}\end{array}\right]}_0^2$
$=-\frac{5}{3}(2\sqrt{2}-0)=\left|\begin{array}{c}\frac{-10\sqrt{2}}{3}\end{array}\right|$
$=\frac{10\sqrt{2}}{3}\Rightarrow \text{Area of bounded Area}=2A=\frac{20\sqrt{20}}{3}$