Unit 6: Integration & Accumulation of Change

The Definite Integral

Defining the definite integral as the limit of a Riemann sum.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Riemann Sums & the Trapezoidal Rule

Area, Exactly, as a Limit

Every Riemann sum so far has been an approximation, built from finitely many rectangles. Before reading on: what would happen to the gap between a Riemann sum and the true area if the number of rectangles grew without bound, each one becoming infinitesimally thin?

Definition — The Definite Integral

The definite integral ∫ₐᵇ f(x) dx is defined as the limit of a Riemann sum as the number of subintervals n → ∞ (equivalently, as the width of each subinterval Δx → 0). It represents the exact net signed area between f's graph and the x-axis on [a, b] — area above the axis counts positively, area below counts negatively.

\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 ∫₋₂² x dx by reasoning about signed area. f(x) = x is a straight line through the origin. On [0, 2], the region is a triangle above the x-axis with area (1/2)(2)(2) = 2, counted positively. On [−2, 0], the identical-sized triangle lies below the x-axis, counted as −2. The net signed area is 2 + (−2) = 0. This matches evaluating an antiderivative directly: [x²/2] from −2 to 2 = 2 − 2 = 0.

Worked Example — Applying Integral Properties

Given ∫₁⁵ f(x) dx = 8 and ∫₁⁵ g(x) dx = −3, find ∫₁⁵ [2f(x) − g(x)] dx. Definite integrals distribute over sums/differences and factor out constants, exactly like limits: 2∫₁⁵f(x)dx − ∫₁⁵g(x)dx = 2(8) − (−3) = 16 + 3 = 19. Separately, given ∫₁⁴f(x)dx = 6 and ∫₄⁷f(x)dx = 5, adjacent intervals combine: ∫₁⁷f(x)dx = 6 + 5 = 11, and reversing the bounds negates the result: ∫₇¹f(x)dx = −11.

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, sketch or picture the region — if the curve dips below the x-axis anywhere in [a, b], that portion subtracts from the total instead of adding to it, and 'area' in the everyday sense (always positive) and the definite integral's signed area can genuinely disagree.

Common Mistakes

Treating a definite integral as always representing a positive, physical area.
A definite integral is signed area — where f(x) < 0, that portion contributes negatively, which is exactly why ∫₋₂² x dx came out to 0 despite the curve clearly enclosing nonzero regions on both sides.
Forgetting that reversing the limits of integration negates the integral's value.
∫ᵦᵃ f(x)dx = −∫ₐᵇ f(x)dx by definition — this isn't optional, and skipping the sign flip when bounds get reordered mid-problem produces the wrong answer.

Key Takeaways

The definite integral is defined as the limit of a Riemann sum as the number of rectangles goes to infinity — the exact net signed area under a curve.
Area below the x-axis counts negatively, so a definite integral can be smaller than, equal to, or even the negative of the region's everyday physical area.
Definite integrals obey linearity (sums and constant multiples) and interval-splitting/reversal properties, all inherited directly from the underlying Riemann sums.

Summary

The definite integral is now defined exactly, but evaluating one from the limit definition every time would be painfully slow. The next lesson connects it directly to antiderivatives — the Fundamental Theorem of Calculus.

Track your progress

← Riemann Sums & the Trapezoidal Rule Back to Unit 6: Integration & Accumulation of Change Applying the Fundamental Theorem→