The area of the region bounded by the line $y=x$ and the curve $y=x^3$ is

• A

  0.2 sq unit
• B

  0.3 sq unit
• C

  0.4 sq unit
• Correct answer
  D

  0.5 sq unit

Solution

$\because y=x$ and $y=x$ The curve intersect at $x=-1, y_i=-1 ; x=0, y=0$ and $x=1, y=1$.
Therefore, area $=-\int_{-1}^0\left(x^3-x\right) d x+\int_0^1\left(x-x^3\right) d x$ $=-\left[\frac{x^4}{4}-\frac{x^2}{2}\right]_{-1}^0+\left[\frac{x^2}{2}-\frac{x^4}{4}\right]_0^1$ $=-\left(\frac{1}{4}-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{4}\right)$ $=-\frac{1}{2}+1=\frac{1}{2}=0.5$ sq. units

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