Hard

Area Between Two Curves

Simple Explanation

The area of the region trapped between two curves f(x) and g(x), where f(x)≥g(x) on [a,b], is found by integrating their difference: ∫ₐᵇ[f(x)−g(x)]dx — subtracting away the area that would otherwise be double-counted underneath the lower curve.

Why Do We Need It?

Many real regions of interest are bounded above and below by two different curves, not by a curve and the x-axis — this generalizes the area-under-a-curve idea to handle that far more common situation.

See It

The region between y = x+2 and y = x², from x = −1 to x = 2

Formula

Area Between Two Curves

Area = ∫ₐᵇ [f(x) − g(x)] dx (where f(x) ≥ g(x) on [a,b])

The area of the region trapped between two curves equals the integral of the "top" function minus the "bottom" function, over the interval where the top function stays on top.

f(x): — the upper (greater-valued) function on [a,b]
g(x): — the lower (lesser-valued) function on [a,b]
a, b: — the x-values where the two curves intersect (or the given boundaries)

When to use it: Whenever the area of a region enclosed between two curves is needed — first find where they intersect, then integrate their difference.

Worked Example

Find the area between two curves

Find the area of the region bounded by f(x) = x+2 and g(x) = x².

Why Does This Work?

∫[f(x)−g(x)]dx can be split into ∫f(x)dx − ∫g(x)dx — the area under the top curve, minus the area under the bottom curve, both measured down to the x-axis — and whatever lies below the x-axis or below g(x) cancels out exactly, leaving only the region genuinely trapped between the two curves.

Real-Life Example

Material saved by a redesigned part

An engineer redesigns a machined part, changing its outline from one curve to another, and needs to know exactly how much material the new design saves.

The saved cross-sectional area is exactly the area between the old outline curve and the new outline curve — computed with this same technique.

Practice

Find the area between f(x) = x and g(x) = x², from x = 0 to x = 1.

Hard

Common mistake

Integrating g(x)−f(x) instead of f(x)−g(x) (subtracting in the wrong order) — this produces a negative value instead of the true, positive area; always confirm which curve is on top first.

Quick Review

Area = ∫ₐᵇ [f(x)−g(x)]dx, where f(x) is the upper curve on [a,b].
First find the intersection points — these are usually the integration bounds.
Test a point inside the interval to confirm which curve is genuinely on top.