Question

Find the area between the curves $y=x\text{and}y=x^2$

Answer 1

Given: Parabola: $y=x^2$ line: $y=x$
Point of intersection of both:

$x^2=x$

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

$\Rightarrow x=0,1$

$\to y=x=0,1$

So, the point of intersections are $(0,0)\&(1,1)$
Drawing the graph


Area required

$={\int }_0^1{y}_{1}dx-{\int }_0^1{y}_{2}dx$

Where $y_1=c\&y_2=x^2$

$={\int }_0^{1}xdx-{\int }_0^1{x}^{2}dx$

$={\int }_0^1(x-x^2)dx$

$={[\frac{x^2}{2}-\frac{x^3}{3}]}_0^1$

$=(\frac{1}{2}-\frac{1}{3})-(0-0)$

$=\frac{1}{6}$

Final answer:
Therefore, required area $=\frac{1}{6}$ square units.