Question

The area bounded by $y={\left(2x\right)}^{1/2}$ and $x={\left(2y\right)}^{1/2}$ is

• A. $\frac{4}{3}$ sq. units
• B. $\frac{13}{2}$ sq. units
• C. $\frac{12}{5}$ sq. units
• D. $\frac{4}{25}$ sq. units

Answer 1

Answer: (A)

We have, $y=\sqrt{2x}$
$\Rightarrow y^2=2x\cdots \left(i\right)$ a parabola with vertex $\left(0,0\right)$.
and $x={\left(2y\right)}^{1/2}$
$\Rightarrow x^2=2y\dots \left(ii\right)$ a parabola with vertex $\left(0,0\right)$.
$\left(i\right)$ and $\left(ii\right)$ intersects at $\left(0,0\right)$ and $\left(2,2\right)$.

Required area = area of shaded region
$=\underset{0}{\overset{2}{\int }}\left(\sqrt{2x}-\frac{x^2}{2}\right)dx$
$=\frac{2}{3}\sqrt{2}{\left|x^{3/2}\right|}_0^2-\frac{1}{2}{\left|\frac{x^3}{3}\right|}_0^2$
$=\frac{2\sqrt{2}}{3}\left(2\sqrt{2}\right)-\frac{1}{2}\times \frac{8}{3}$
$=\frac{8}{3}-\frac{4}{3}$
$=\frac{4}{3}$ sq. units