Mathematics - Area Under Curves Question with Solution | TestHub

Given curves are $y=x^2$ and $x=y^2$, which is the form of parabola.The point of intersection, $x={\left(x^2\right)}^2$$\begin{array}{l}\Rightarrow x=x^4\Rightarrow x\left(1-x^3\right)=0\\ \Rightarrow x=0\text{ and }1=x^3\\ \Rightarrow x=0\text{ and }x=1\\ \text{ When }x=0,\text{ then }y=0\\ \text{ When }x=1,\text{ then }y=1^2=1\end{array}$ $\therefore$ The point of intersection is (0,0) and (1,1) $\therefore$ Area of shaded region $$\begin{array}{l}={\int }_0^1\left(y_2-y_1\right)dx\\ ={\int }_0^1\left[\sqrt{x}-x^2\right]dx={\left[\frac{x^{3/2}}{3/2}-\frac{x^3}{3}\right]}_0^1\\ =\frac{2}{3}(1{)}^{3/2}-\frac{(1{)}^3}{3}-0-0\\ =\frac{2}{3}-\frac{1}{3}=\frac{1}{3}\mathrm{sq}\text{ units }\end{array}$$