Area between curves

The definite integral measures the area between a curve and the x-axis. Swap that axis for a second curve and the same idea measures the area of the region trapped between them.

Top minus bottom

Picture a thin vertical strip inside the region. Its width is dx and its height is the distance from the upper curve down to the lower one — that is, f(x) − g(x). Add up the strips across the interval where the region lives:

Area = ∫ₐᵇ [ f(x) − g(x) ] dx

where f is the top curve, g is the bottom, and a and b are the x-values where they cross.

The recipe

1. Find the intersection points by setting f(x) = g(x); these are your limits a and b.
2. Decide which curve is on top across that interval (the taller one).
3. Integrate the difference (top − bottom).

Worked example

Find the area between f(x) = 3.2 − 0.25x² (top) and g(x) = 0.15x² + 0.4 (bottom).

• Intersect: 3.2 − 0.25x² = 0.15x² + 0.4 → 2.8 = 0.4x² → x² = 7 → x = ±√7.
• Top − bottom: (3.2 − 0.25x²) − (0.15x² + 0.4) = 2.8 − 0.4x².
• Integrate from −√7 to √7: ∫ (2.8 − 0.4x²) dx = [ 2.8x − 0.4x³/3 ].

Evaluating from −√7 to √7 gives the shaded area — about 9.9 square units.

Why the "top − bottom" always works

Because it measures a height, f(x) − g(x) is correct even when the region dips below the x-axis: both curves shift by the same amount, so their difference — and the area — is unchanged. You never have to worry about signs, just which curve is higher.

Why this matters

"Difference of two accumulations" is a common real quantity: the extra distance one runner covers over another (area between their velocity curves), or the consumer surplus between a demand curve and a price line. Area between curves is how you measure a gap that changes continuously.