Question

Find the area of the region bounded by $y=\sqrt{x}andy=x.$

Answer 1

Given equation of curves are $y=\sqrt{x}andy=x.$

$\Rightarrow x=\sqrt{x}\Rightarrow x^2=x$

$\Rightarrow x^2-x=0\Rightarrow x(x-1)=0$

$\Rightarrow x=0,1$

$\therefore$ Required area of shaded region, $A={\int }_0^1\left(\sqrt{x}\right)dx-{\int }_0^{1}xdx={\left[2.\frac{x^{\frac{3}{2}}}{3}\right]}_0^1-{\left[\frac{x^2}{2}\right]}_0^1=\frac{2}{3}.1-\frac{1}{2}=\frac{2}{3}-\frac{1}{2}=\frac{1}{6}squnits$