Math · Calculus I · Concept

Area between two curves

To find the area between two curves, find where they intersect, decide which curve is on top, and integrate top minus bottom from a to b: A = ∫(f(x) − g(x)) dx, where f(x) ≥ g(x) on [a, b]. If the curves cross inside the interval, split the integral at each crossing so every piece is positive.

Top minus bottom

A thin vertical strip at x has height f(x) − g(x), where f is the upper curve, and width dx. Adding the strips gives the area. Only the difference matters, so the formula works even where both curves dip below the x-axis.

A = \int_{a}^{b} (f (x) - g (x)) d x, f (x) \geq g (x)

\begin{array}{c} A = \int_{a}^{b} (f (x) - g (x)) d x \\ f (x) \geq g (x) on [a, b] \end{array}

Find the limits where the curves meet

When the curves enclose a region, its left and right edges are the x-values where they meet. Solve f(x) = g(x); factoring or the quadratic formula usually finishes the job.

Decide which curve is on top

Test one x-value between the intersections, or look at a graph. Subtracting in the wrong order gives the negative of the area.

Split where the curves cross

If the curves swap places inside [a, b], split the interval at each crossing and integrate top minus bottom on each piece. That is the same as integrating the absolute value of the difference. Integrating f − g straight across a crossing lets the pieces cancel and gives a net difference, not an area.

A = \int_{a}^{b} | f (x) - g (x) | d x

Strips in the other direction

When a region is bounded on the left and right by curves x = h(y), use horizontal strips and integrate right minus left with respect to y. This avoids splitting when a vertical strip would meet different curves in different places.

Common mistakes

Subtracting in the wrong order, which gives a negative area.
Integrating f − g across a crossing point, so positive and negative pieces cancel.
Using the x-axis as a boundary when the region lies between two curves: the axis plays no part unless it is one of the curves.
Solving f(x) = g(x) and keeping only one intersection, so the interval is wrong.

Key terms

Area between curves: The area of a region bounded by two graphs, found by integrating the upper curve minus the lower curve: A = ∫ₐᵇ (f(x) − g(x)) dx where f ≥ g. Where the curves cross, the interval is split so each piece is positive.
Definite integral: ∫ₐᵇ f(x) dx: the signed area between the graph of f and the x-axis from a to b, defined as the limit of Riemann sums. Area below the axis counts as negative.
Signed area: Area counted with a sign: regions above the x-axis count as positive and regions below as negative, so they can cancel. Total area counts every region as positive.
Antiderivative: A function whose derivative is the given function, such as x³ for 3x². Any two antiderivatives on one interval differ by a constant, which is why answers carry + C.

Work through an example

Find the area of the region enclosed by y = 4 − x² and y = x + 2.

Find the area between a parabola and a line →

Split the area where two curves cross →

Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.

OpenStax Calculus Volume 1 — Areas between curves

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See both curves in Graph Compare rectangle sums with the exact area Check the work in Math Open worked example on a board Area between curves in Math Reference

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