Area between a parabola and a line

Problem

Compute the area of the region enclosed between $y=x^2$ and $y=x$.

Solution

Intersections: $x^2 = x \iff x(x-1)=0$, that is $x=0$ and $x=1$. On $[0,1]$ the line $y=x$ lies above the parabola $y=x^2$, so: $A = \int_0^1(x-x^2)\,dx = \left[\frac{x^2}{2}-\frac{x^3}{3}\right]_0^1 = \frac{1}{2}-\frac{1}{3} = \boxed{\dfrac{1}{6}}.$

Links

Topics: Integral
Concepts: Area between two curves · Definite integral
Skills: Analytic geometry · Integrate
Exercise types: Integral computation