Question

The area bounded by the parabolas $y = 4x^{2}, y = \frac{x^{2}}{9}$ and the 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. $\frac{20\sqrt{2}}{3}$ sq. units

Answer 1

Answer: (D)

$ y =4 x^{2} \,\,\,...(i)$
$ y =\frac{x^{2}}{4} \,\,\,...(ii)$
$\therefore A=\int\limits_{0}^{2}\left[\frac{\sqrt{y}}{2}-3 \sqrt{y}\right] d y$
$=\left(\frac{1}{2}-3\right) \int\limits_{0}^{2} \sqrt{y} \,d y$
$=\left(\frac{-5}{2}\right) \cdot \frac{2}{3}\left[y^{\frac{3}{2}}\right]_{0}^{2}$
$=-\frac{5}{3}(2 \sqrt{2}-0)=\left|\frac{-10 \sqrt{2}}{3}\right|$
$=\frac{10 \sqrt{2}}{3} $
$\Rightarrow $ Area of bounded Area $=2 A=\frac{20 \sqrt{2}}{3}$