Integrals

The Definite Integral

Defining the definite integral as a limit of Riemann sums.

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

Prerequisites

Riemann Sums

Area, Exactly, as a Limit

Definition — The Definite Integral

∫ₐᵇ f(x) dx is defined as the limit of a Riemann sum as the number of subintervals n → ∞. It represents net signed area — positive above the x-axis, negative below.

\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i)\, \Delta x

Worked Example — Interpreting Net Signed Area

Evaluate ∫₁⁵ (x − 3) dx. f(x) = x − 3 crosses zero at x = 3. On [1,3], f is negative, forming a triangle below the axis with legs 2 and 2 — area 2, counted as −2. On [3,5], f is positive, an identical triangle above the axis — area 2, counted as +2. Net: −2 + 2 = 0. Confirmed algebraically: [x²/2 − 3x]₁⁵ = (12.5−15) − (0.5−3) = −2.5 − (−2.5) = 0.

Worked Example — Applying Integral Properties

Given ∫₂⁶ f(x) dx = 10 and ∫₂⁶ g(x) dx = 4, find ∫₂⁶ [3f(x) − 2g(x)] dx. By linearity: 3(10) − 2(4) = 30 − 8 = 22.

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

Before computing a definite integral, check whether the curve dips below the x-axis anywhere in the interval — that portion subtracts from the total instead of adding to it.

Common Mistakes

Treating a definite integral as always a positive, physical area.
A definite integral is signed area — area below the x-axis contributes negatively, exactly why ∫₁⁵(x−3)dx came out to 0.

Key Takeaways

The definite integral is the exact limit of a Riemann sum — net signed area, not always a positive physical area.
Definite integrals obey linearity, inherited directly from the underlying Riemann sums.

Summary

The definite integral is now defined exactly. The next lesson connects it to a faster tool — antiderivatives.

Track your progress

← Riemann Sums Back to Integrals Antiderivatives→