Area Between Curves

When two curves cross or bound a region on the coordinate plane, we can calculate the exact area trapped between them using definite integration. This powerful application of calculus extends the basic idea of finding area under a single curve to finding the area sandwiched between an upper curve and a lower curve over some interval. The core insight is simple: if you know the area under the top curve and the area under the bottom curve (both measured from the x-axis), then subtracting the second from the first gives exactly the area of the region between them. This works because integration is linear and additive, so the difference of two integrals equals the integral of the difference. Area between curves appears throughout science and economics — for example, calculating the work done by a varying force versus a reference force, finding the consumer surplus in a supply-and-demand model, or determining the cross-sectional area of an engineering component. Mastering this concept is essential preparation for volumes of revolution, physics applications, and multivariable calculus.

Key takeaways
  • Identify the upper and lower functions
  • Find the limits of integration
  • Set up and evaluate the integral
  • Using horizontal slices (integrating with respect to y)

Identify the upper and lower functions

Determine which function is on top (larger y-values) and which is on the bottom over the interval of interest. If f(x) ≥ g(x) on [a, b], then f is the upper curve and g is the lower curve. If the curves cross within the interval, you must split the integral at each intersection point and flip which function is 'upper' in each sub-interval.

Find the limits of integration

The limits a and b are either given explicitly or found by solving f(x) = g(x) to locate intersection points. Set the two expressions equal, solve for x, and verify which curve lies above the other between those x-values by testing a sample point.

Set up and evaluate the integral

Write the area as A = ∫[a to b] [f(x) − g(x)] dx. Expand or simplify the integrand, then antidifferentiate term by term. Evaluate at the upper limit minus the lower limit. The result must be non-negative; if you get a negative value, check which curve is actually on top.

Using horizontal slices (integrating with respect to y)

Sometimes it is easier to integrate with respect to y instead of x — especially when curves are expressed as x = f(y) and x = g(y), or when the region is awkward to split along the x-axis. The formula becomes A = ∫[c to d] [x_right(y) − x_left(y)] dy, where c and d are the y-coordinates of the boundary intersections.

The Key Equation

A=∫ab[f(x)−g(x)] dx,f(x)≥g(x) on [a,b]A = \int_{a}^{b} \bigl[f(x) - g(x)\bigr]\, dx, \quad f(x) \geq g(x) \text{ on } [a,b]
  • A — Area (square units): The positive area of the region bounded between the two curves.
  • f(x) — Upper function (y-units): The curve with larger y-values on the interval [a, b].
  • g(x) — Lower function (y-units): The curve with smaller y-values on the interval [a, b].
  • a,\, b — Limits of integration (x-units): The left and right x-coordinates bounding the region, often found from f(x) = g(x).

Worked Example

Problem. Find the area of the region enclosed between f(x) = x + 2 and g(x) = x² (the parabola and the line).

A=92 square unitsA = \dfrac{9}{2} \text{ square units}

Answer. The area between the curves is 9/2 square units.

Figure
Diagram
xy0−12(−1,1)(2,4)f(x)=x+2g(x)=x²Area = 9/2
Origins · 1 minGottfried Wilhelm Leibniz and how this was discovered
GW
Gottfried Wilhelm Leibniz
1646–1716 · German

Leibniz independently developed calculus alongside Newton, introducing the elegant notation ∫ and dx that we still use today. His systematic approach to integration as a summation process — the word 'calculus' itself coming from the Latin for 'small stone' — directly enables the area-between-curves formula. His notation makes the difference formula A = ∫[a to b][f(x)−g(x)]dx immediately readable and meaningful. Leibniz invented one of the first mechanical calculators capable of multiplication and division, which he called the 'Stepped Reckoner.' He carried it to London to demonstrate to the Royal Society in 1673, a full decade before his famous calculus papers appeared.

  • The Definite Integral — Area between curves is computed using definite integrals; a solid understanding of ∫[a to b] f(x) dx is prerequisite.
  • Fundamental Theorem of Calculus — The FTC is what allows us to evaluate definite integrals by antidifferentiation, making area calculations tractable.
  • Volumes of Revolution — Rotating a region between curves around an axis uses the same setup as area between curves, extended to 3D.
  • Riemann Sums — The area between curves can be understood as the limit of summing thin rectangular strips, connecting intuition to the formal integral.