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

Graphical or geometric meaning

A vertical slice begins on the parabola and ends on the line, giving height $2x-x^2$. That gap vanishes at both intersections and peaks between them, matching the lens-shaped region in the sketch.

Common mistakes and why they fail

Verification and reasonableness checks

• Factor the gap to verify its sign between bounds.
• Estimate using a bounding rectangle.
• Confirm the answer is nonnegative and has squared units.

Let geometry choose the slicing direction

With vertical slices, area is $\int(\text{top}-\text{bottom})\,dx$; with horizontal slices, it is $\int(\text{right}-\text{left})\,dy$. Find intersections and test which curve supplies each boundary between them. If the order changes, split the integral instead of allowing negative pieces to cancel. Horizontal slices may avoid several vertical subregions, so solve for $x$ before committing to $dx$. A sketch identifies the enclosed region and prevents integrating between unrelated intersections. The final area must be nonnegative and should fit inside an obvious bounding rectangle. If it exceeds that rectangle, revisit the curve order, bounds, or antiderivative rather than accepting a formally simplified expression.

Practice

1. Find the area between $y=x$ and $y=0$ on $[0,3]$.
2. Where do $y=x^2$ and $y=4$ meet?
3. What is a horizontal slice width?

Connections and next steps

• Area Under a Curve
• Applications of Integration
• Volumes by Disks and Washers