Question

Find the area of the region bounded by the parabola $y=x^2$ and the line $y=x$ in the first quadrant.

Answer 1

$y=x^2,y=x$

on solving, $x-x^2=0$

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

$\Rightarrow x=0,x=1$

${\int }_0^1(x-x^2)dx=$ $[\frac{x^2}{2}-\frac{x^3}{3}{]}_0^1$

$=\frac{1}{2}-\frac{1}{3}+0=\frac{3-2}{6}=\frac{1}{6}$