Question

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

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

Answer 1

The correct option is D $\frac{20\sqrt{2}}{3}$

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

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 first quadrant.

$\Rightarrow$ Required area $=2{\int }_0^2(3\sqrt{y}-\frac{\sqrt{y}}{2})dy$

$=2{\int }_0^2\left(\frac{6\sqrt{y}-\sqrt{y}}{2}\right)dy$

$=2{\int }_0^2\left(\frac{5\sqrt{y}}{2}\right)dy$

$=5{\left(\frac{y^{\frac{3}{2}}}{3/2}\right)}_0^2$

$=\frac{10}{3}(2^{\frac{3}{2}}-0)$

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