Summary of Areas Between Curves

Just as definite integrals can be used to find the area under a curve, they can also be used to find the area between two curves.
To find the area between two curves defined by functions, integrate the difference of the functions.
If the graphs of the functions cross, or if the region is complex, use the absolute value of the difference of the functions. In this case, it may be necessary to evaluate two or more integrals and add the results to find the area of the region.
Sometimes it can be easier to integrate with respect to [latex]y[/latex] to find the area. The principles are the same regardless of which variable is used as the variable of integration.

Key Equations

Area between two curves, integrating on the [latex]x[/latex]-axis
[latex]A={\displaystyle\int }_{a}^{b}\left[f(x)-g(x)\right]dx[/latex]
Area between two curves, integrating on the [latex]y[/latex]-axis
[latex]A={\displaystyle\int }_{c}^{d}\left[u(y)-v(y)\right]dy[/latex]