The area of region bounded by two curves $y^2=x$ and $x^2=y$ is:

$\frac{1}{3}$ sq. unit

The correct answer is Option (3) → $\frac{1}{3}$ sq. unit

Given equations,

$y^2=x$   ...(1)

$x^2=y$   ...(2)

From (1) and (2), we can get

$(y^2)^2=y$

$⇒y(y^3-1)=0$

$⇒y(y-1)(y^2+y+1)=0$

$⇒y=0\,or\,1$

$∴Area=\left|\int\limits_0^1(\text{Right-Left})dy\right|$

$=\left|\int\limits_0^1y^2\,dy-\int\limits_0^1\sqrt{y}\,dy\right|$

$=\left|\frac{1}{3}-\frac{2}{3}\right|=\frac{1}{3}$ sq. unit