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

• A

  $\frac{1}{2}$ sq. units
• B

  $\frac{1}{3}$ sq. units
• C

  $\frac{2}{3}$ sq. units
• Correct answer
  D

  $\frac{1}{6}$ sq. units

Solution

The required area is shaded The point of intersection of the curves are $x^2=x \Rightarrow x(x-1)=0$ i.e. $O(0,0)$ and $P(1,1)$
$$ \begin{aligned} & \therefore A=\int_0^1\left(x-x^2\right) d x \\ & =\left[\frac{x^2}{2}\right]_0^1-\left[\frac{x^3}{3}\right]_0^1=\frac{1}{2}-\frac{1}{3}=\frac{1}{6} \end{aligned} $$

Hence, the correct answer is option: 'D'.