Find the area of the region enclosed between the curves $y=x$ and $y=x^2$ from $x=0$ to $x=1$.

Step-by-step solution

1. Set up the integral for the area as $A={\int }_0^1(\text{upper curve}-\text{lower curve})dx$.
2. For $0\le x\le 1$, $y=x$ lies above $y=x^2$, so $A={\int }_0^1(x-x^2)dx$.
3. Integrate termwise: ${\int }_0^{1}xdx={\left[\frac{x^2}{2}\right]}_0^1=\frac{1}{2}$ and ${\int }_0^1{x}^{2}dx={\left[\frac{x^3}{3}\right]}_0^1=\frac{1}{3}$.
4. Subtract the results: $A=\frac{1}{2}-\frac{1}{3}=\frac{3-2}{6}=\frac{1}{6}$.
5. State the final answer.

Concept tested: Area between two curves

Published 8 September 2026 · How this solution was prepared and checked