Area Between Curves

The central idea

For vertical slices, area between an upper curve $y=f(x)$ and lower curve $y=g(x)$ on $[a,b]$ is $A=\int_a^b[f(x)-g(x)]dx$, provided $f\ge g$. Horizontal slices use $A=\int_c^d[x_{\text{right}}(y)-x_{\text{left}}(y)]dy$. Intersections set bounds and may require splitting when order changes.

Definitions, hypotheses, and notation

An absolute-value expression $\int|f-g|dx$ is conceptually correct, but an exact evaluation still requires locating sign changes and removing the absolute value piecewise. If vertical slices demand multiple formulas while horizontal slices use one, changing orientation reduces both algebra and error risk. The same region must give the same area either way.

Unbounded regions require improper integrals, so finite-looking intersections alone do not guarantee finite area. For ordinary bounded regions, a slice gap should go to zero at boundary intersections and remain nonnegative inside; these are quick setup checks before antiderivatives are computed.

Conceptual meaning

Each thin rectangle has thickness and a nonnegative gap between boundaries. The integral accumulates these gaps. Choosing vertical or horizontal slices is a modeling decision: use the orientation that describes the region with fewer pieces.

A dependable method and decision rule

1. Sketch both curves and solve their intersection equations.
2. Choose vertical or horizontal slices.
3. Identify top minus bottom or right minus left on each interval.
4. Split wherever boundary identity or order changes.
5. Integrate and ensure every piece contributes nonnegative area.

Fully worked example

Common mistakes and why they fail