Question

The area between the parabola $y=x^2$ and the line $y=x$ is

• A. $\frac{1}{6}$ sq. units
• B. $\frac{1}{3}$ sq. units
• C. $\frac{1}{2}$ sq. units
• D. None of these

Answer 1

The correct option is A $\frac{1}{6}$ sq. units

For points of intersection of the equations $y=x^2$ and $y=x$
$\therefore x^2-x=0\Rightarrow x(x-1)=0$
$x=0$ or $x=1\Rightarrow Y=0$ or $y=1$
Hence, the coordinates of their points of intersection are O(0, 0) and P (1, 1).
$\therefore$ Required area (shaded region)
$={\int }_0^1(x-x^2)dx={[\frac{x^2}{2}-\frac{x^3}{3}]}_0^1$
$=[(\frac{1}{2}-\frac{1}{3})-0]=\frac{1}{6}$ sq. units