area between curves

The region trapped between two graphs, found by integrating the difference of the top and bottom functions.

The area between curves $y=f(x)$ (on top) and $y=g(x)$ (on bottom) over $[a,b]$ is $\int_a^b [f(x)-g(x)]\,dx$ — the definite integral of the vertical gap between them, which must stay non-negative wherever the "top" function is correctly identified.

Examples

The area between $y=x$ and $y=x^2$ on $[0,1]$ is $\int_0^1(x-x^2)\,dx = \tfrac16$ .