Definite Integral as a Limit

The definite integral is one of the most powerful ideas in mathematics, but at its heart it rests on a beautifully simple question: how do you find the exact area under a curve? The answer lies in a limiting process — you approximate the area with many thin rectangles, then take the limit as those rectangles become infinitely thin. This process is called a Riemann sum, and when it converges to a single value, that value is the definite integral. Formally, the definite integral of a function f(x) from x = a to x = b is defined as the limit of a Riemann sum as the number of subintervals n approaches infinity and each subinterval's width approaches zero. This definition transforms the intuitive notion of 'area under a curve' into a precise mathematical object that can be computed, compared, and applied across physics, engineering, economics, and beyond. Understanding the definite integral as a limit of sums is essential not just for computation, but for interpreting what integration actually means. When you see an integral, you should visualise an infinite accumulation of infinitely small contributions — a perspective that unlocks deep connections between rates of change and accumulated quantities, ultimately leading to the Fundamental Theorem of Calculus.

Key takeaways

Partition the Interval
Choose Sample Points
Form the Riemann Sum
Take the Limit
Convergence and Integrability

Partition the Interval

Start with a continuous function f(x) on a closed interval [a, b]. Divide this interval into n equal subintervals, each of width Δx = (b − a)/n. The endpoints of these subintervals are x₀ = a, x₁ = a + Δx, x₂ = a + 2Δx, …, xₙ = b. You have just created a partition of [a, b].

Choose Sample Points

In each subinterval [xᵢ₋₁, xᵢ], choose a sample point xᵢ. Common choices include the left endpoint (xᵢ = xᵢ₋₁), the right endpoint (xᵢ = xᵢ), or the midpoint. The choice affects the approximation but not the limit. For the right-endpoint rule: xᵢ = a + i·Δx.

Form the Riemann Sum

Approximate the area under f(x) on each subinterval by the area of a rectangle with height f(xᵢ) and width Δx. Sum all these rectangles: Sₙ = Σᵢ₌₁ⁿ f(xᵢ)·Δx. This is the Riemann sum. For n = 4 rectangles under f(x) = x² on [0, 2], using right endpoints gives Δx = 0.5, and S₄ = f(0.5)(0.5) + f(1)(0.5) + f(1.5)(0.5) + f(2)(0.5) = 0.5(0.25 + 1 + 2.25 + 4) = 0.5(7.5) = 3.75.

Take the Limit

As n → ∞, the rectangles become infinitely thin and their sum approaches the exact area. The definite integral is defined as this limit: ∫ₐᵇ f(x) dx = lim(n→∞) Σᵢ₌₁ⁿ f(xᵢ*)·Δx. For f(x) = x² on [0, 2], the limit gives exactly 8/3 ≈ 2.667. Notice that our rough approximation of 3.75 with only 4 rectangles overestimates the true value — as n increases, the Riemann sums converge to 8/3.

Convergence and Integrability

A function is called Riemann integrable on [a, b] if this limit exists and is the same regardless of which sample points xᵢ* are chosen. All continuous functions on closed intervals are Riemann integrable. Even some discontinuous functions (like step functions) are integrable, as long as the set of discontinuities is not too large. The Riemann sum limit is the rigorous foundation on which all of integration theory is built.

The Key Equation

\int_{a}^{b} f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f\!\left(a + i\cdot\frac{b-a}{n}\right)\cdot\frac{b-a}{n}

\int_{a}^{b} — Definite integral (depends on context): The integral sign with lower limit a and upper limit b; represents the accumulated signed area between f(x) and the x-axis from x = a to x = b.
f(x) — Integrand (depends on context): The function being integrated; its values serve as the heights of the approximating rectangles.
dx — Differential (same as x): The infinitesimal width element; represents the limiting value of Δx = (b − a)/n as n → ∞.
n — Number of subintervals (dimensionless): The count of equal partitions of [a, b]; as n → ∞, the approximation becomes exact.
\frac{b-a}{n} — Subinterval width (Δx) (same as x): The width of each rectangle in the uniform partition; shrinks toward zero as n grows.

Worked Example

Problem. Use the limit definition of the definite integral (right-endpoint Riemann sum) to evaluate ∫₀³ (2x) dx. Verify your answer using the area formula for a triangle.

\int_{0}^{3} 2x\,dx = \lim_{n\to\infty} 9\!\left(1+\frac{1}{n}\right) = 9

Answer. The definite integral equals 9.

Figure

Diagram

Origins · 1 minBernhard Riemann and how this was discovered

Bernhard Riemann

1826–1866 · German

Georg Friedrich Bernhard Riemann gave the first rigorous definition of the integral as a limit of sums in his 1854 habilitation thesis. His formulation — now called the Riemann integral — precisely defined what it means for a function to be integrable and established the conditions under which the limit of Riemann sums exists. This put the intuitive ideas of Newton and Leibniz on a completely rigorous footing and became the standard definition of integration taught in analysis courses worldwide. Riemann died of tuberculosis at only 39, yet his short life produced some of the most influential mathematics in history — including the Riemann Hypothesis, one of the seven Millennium Prize Problems, which remains unsolved to this day and carries a $1,000,000 prize.

Riemann Sums and Approximation — Riemann sums are the direct building blocks of the limit definition; mastering left, right, and midpoint sums prepares you for understanding what happens as n → ∞.
Fundamental Theorem of Calculus — The FTC provides a practical way to evaluate definite integrals using antiderivatives, bypassing the limit definition — but it is only valid because the limit definition exists.
Sigma Notation and Finite Series — Riemann sums are written in sigma notation; knowing closed-form formulas like Σi = n(n+1)/2 is essential for evaluating limits of Riemann sums algebraically.
Net Signed Area and the Definite Integral — The limit definition naturally produces signed areas — rectangles below the x-axis contribute negative values. Understanding signed area extends the accumulation concept to functions that cross the x-axis.