Question

Find the area between the curves y = x and $y=x^2$.

Answer 1

Given, curve (represents an upward parabola with vertex (0, 0))

$x^2=y$ ...(i)

and equation of the line y = x ...(ii)

For intersection point $x^2=x$

[From Eqs. (i) and (ii)]

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

When x = 0, y = 0 and when x = 1, y = 1

Thus, point of intersection of parabola and line are (0, 0) and (1, 1).

$\therefore$ Required area $={\int }_0^1(y_2-y_1)dx={\int }_0^1(x-x^2)dx$

$=[\frac{x^2}{2}-\frac{x^3}{3}{]}_0^1=(\frac{1}{2}-\frac{1}{2})-0=\frac{1}{6}squnit$