Question

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

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

Answer 1

The correct option is C

$\frac{20\sqrt{2}}{3}$

Explanation for the correct option:

Finding the area:

The given two parabolas are $x^2=\frac{y}{4}\text{and}{x}^2=9y$

Thus, $x=\frac{\sqrt{y}}{2}\text{and}x=3\sqrt{y}$

We have to find the area bounded between the parabolas and the straight line $y=2$.

The required area is equal to the shaded region in the drawn figure. The area of the shaded region is twice the area shaded in the first quadrant.

$\begin{array}{rcl}\therefore \text{Required area}& =& 2{\int }_0^2\left(3\sqrt{y}-\frac{\sqrt{y}}{2}\right)dy\\ & =& 2{\int }_0^2\left(\frac{6\sqrt{y}-\sqrt{y}}{2}\right)dy\\ & =& 2{\int }_0^2\left(\frac{5\sqrt{y}}{2}\right)dy\\ & =& 5{\left(\frac{y^{3/2}}{3/2}\right)}_0^2\\ & =& \frac{10}{3}\left(2^{3/2}-0\right)\\ & =& \frac{20\sqrt{2}}{3}\end{array}$

Therefore, the area bounded between the parabolas $x^2=\frac{y}{4}and{x}^2=9y$ and the straight line $y=2$ is $\frac{20\sqrt{2}}{3}$

Hence, option (C) is the correct answer.