Area Between Curves

Measuring Enclosed Regions

A natural application of the definite integral is the area between two curves. If $f(x) \ge g(x)$ on $[a, b]$, the area of the region trapped between them is $$A = \int_a^b \big(f(x) - g(x)\big)\, dx.$$ The idea generalizes the "area under a curve": each thin vertical strip has height (top curve minus bottom curve) and width $dx$, and integrating adds them all up. The key is always top minus bottom — the upper curve's function minus the lower one's, so the height is nonnegative.

The method:

1. Find the intersection points — solve $f(x) = g(x)$ to get the limits $a$ and $b$ where the region begins and ends.
2. Determine which curve is on top over each interval (they may swap; if so, split the integral at the crossing and take top-minus-bottom on each piece).
3. Integrate the difference.

Because you subtract $g$ from $f$, the answer is a genuine geometric area (always positive), even if part of the region lies below the $x$-axis — the axis is irrelevant when measuring between two curves. This is a crucial difference from a single definite integral (which is signed): between-curves area uses the difference of the functions, sidestepping sign issues.

Sometimes it is easier to integrate with respect to $y$ (horizontal strips, right curve minus left) — a useful alternative when the curves are naturally functions of $y$. Either way, the principle is the same: integrate the width of the region across its extent.

Common pitfall: integrating $f - g$ without checking which curve is on top, or forgetting that the curves may cross within the interval. The height must be top minus bottom (nonnegative); if you subtract in the wrong order you get a negative "area," and if the curves swap places inside $[a, b]$ you must split the integral at the crossing point and take top-minus-bottom on each piece separately.

Two curves crossing to enclose a lens-shaped region filled with accent vertical strips, each strip labeled top-minus-bottom, summing to the enclosed area.

$$A = \int_a^b \big(f(x) - g(x)\big)\, dx$$