Question

Area enclosed by $y=x^2$ and $x=y^2$

Answer 1

$\frac{1}{3}$

Answer 2

Find intersection points of $y=x^2$ and $x=y^2$ by solving simultaneously, yielding $(0,0)$ and $(1,1)$. In the interval $[0,1]$, $y=\sqrt{x}$ (from $x=y^2$) is the upper curve and $y=x^2$ is the lower curve. The area is computed by integrating the difference between the upper and lower curves from $x=0$ to $x=1$: $\int_0^1 (\sqrt{x} - x^2) dx$. Evaluating this definite integral gives the area.