Applications of Integrals

Area Between Curves

Using a definite integral to find the area between two curves.

Advanced20 min lesson2 min readUpdated August 12, 2026Author not yet attributed

The Gap Between Two Curves

Definition — Area Between Two Curves

The area enclosed between y = f(x) and y = g(x), where f(x) ≥ g(x) on [a, b], is ∫ₐᵇ[f(x) − g(x)]dx — the bounds are usually the curves' intersection points.

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

Worked Example — Finding the Area Between Two Curves

Find the area enclosed between y = x² and y = 2x + 3. Intersections: x² = 2x + 3, so x² − 2x − 3 = 0, which factors as (x−3)(x+1) = 0, giving x = −1, 3. Testing x = 0: the line gives 3, the parabola gives 0 — the line is on top. Area = ∫₋₁³[(2x+3) − x²]dx. Antiderivative: x² + 3x − x³/3. At x = 3: 9 + 9 − 9 = 9. At x = −1: 1 − 3 + 1/3 = −5/3. Area = 9 − (−5/3) = 32/3 ≈ 10.67.

Integral Visualizer

f(x) =

x^2

Integration bounds

a — lower bound0

b — upper bound2

Animate b (the play button above) to watch the shaded area — and the accumulation curve below — grow.

Approximation method

Method

Subdivisions (n)10

Show approximationShow exact area

Trace mode

f(x), with the region between a and b shaded

Show accumulation function A(x)

Accumulation function A(x) = ∫ₐˣ f(t) dt — linked to the graph above; panning or zooming either moves both

What the Fundamental Theorem of Calculus tells us

A(x) = ∫ from a to x of f(t) dt is the accumulation function. At x = b ≈ 2, A(b) ≈ 2.67 — the signed area under f from a ≈ 0 to b.
The Fundamental Theorem of Calculus says A'(x) = f(x): differentiating the accumulation function gives the original function back.
Confirmed numerically here: differentiating the accumulation curve gives A'(b) ≈ 4, matching f(b) ≈ 4.

Numerical approximation comparison

Every method's estimate of ∫ₐᵇ f(x) dx at n = 10 subdivisions, against a high-resolution reference — all should converge toward the same value.
Method	Estimate
Left Riemann sum	2.28
Right Riemann sum	3.08
Midpoint Riemann sum	2.66
Trapezoidal Rule	2.68
Simpson's Rule	2.66667
Exact (high-resolution reference)	2.66667

Analysis of f(x)

y-intercept: (0, 0)
x-intercepts: (0, 0)
Extrema: local min at (0, 0)
Inflection points: none found in view
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [0, 100]

Tip

Always confirm which curve is actually on top with a single test point inside the interval — subtracting in the wrong order produces the exact negative of the correct area.

Common Mistakes

Using the original functions' natural domain endpoints as the bounds instead of their intersection points.
The area between two curves is only well-defined between the x-values where they actually cross — using arbitrary bounds integrates over a region where the 'between' relationship no longer holds.

Key Takeaways

Area between two curves is ∫ₐᵇ[top(x) − bottom(x)]dx, with bounds usually found from the curves' intersection points.
A test point between the intersection points confirms which curve is on top before setting up the integral.

Summary

The final lesson interprets a definite integral more broadly — as the accumulated change in any quantity, not just a geometric area.

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