Riemann Sums

A Riemann sum is a method for approximating the total area under a curve on a graph by dividing the region into a finite number of vertical rectangles, computing the area of each rectangle, and summing them together. This technique forms the conceptual foundation of the definite integral, bridging the gap between discrete sums and continuous accumulation. Named after the German mathematician Bernhard Riemann, these sums allow us to quantify quantities like distance traveled, accumulated rainfall, or work done by a variable force—all situations where we need to 'add up' infinitely many infinitesimally small contributions. The key idea is partition: we split the interval [a, b] over which we want to integrate into n subintervals, choose a sample point inside each subinterval, evaluate the function at that point, multiply by the width of the subinterval, and sum all such products. Different choices of sample point yield different types of Riemann sums—left, right, midpoint, and trapezoidal—each with different levels of accuracy. As the number of subintervals n increases (and their widths shrink toward zero), the Riemann sum converges to the exact value of the definite integral, provided the function is integrable on that interval. Understanding Riemann sums is essential not just as a historical curiosity, but as a practical tool for numerical integration when antiderivatives are impossible or impractical to compute. Techniques such as Simpson's Rule and the Trapezoidal Rule are direct refinements of the Riemann sum idea. In AP Calculus and university-level analysis, Riemann sums serve as the rigorous definition of the definite integral, making them a cornerstone concept for any serious study of calculus.

Key takeaways
  • Define the Partition
  • Choose Sample Points
  • Compute Rectangle Areas
  • Sum All Rectangles
  • Assess Accuracy

Define the Partition

Choose the interval [a, b] and divide it into n equal subintervals. Each subinterval has width Δx = (b − a)/n. The partition points are x₀ = a, x₁ = a + Δx, x₂ = a + 2Δx, …, xₙ = b. For example, integrating over [1, 5] with n = 4 gives Δx = (5−1)/4 = 1, and partition points 1, 2, 3, 4, 5.

Choose Sample Points

In each subinterval [xᵢ₋₁, xᵢ], pick a sample point xᵢ to evaluate the function. Left Riemann Sum: xᵢ = xᵢ₋₁ (left endpoint). Right Riemann Sum: xᵢ = xᵢ (right endpoint). Midpoint Riemann Sum: xᵢ = (xᵢ₋₁ + xᵢ)/2. The choice affects accuracy—for a strictly increasing function, the left sum underestimates and the right sum overestimates the true integral.

Compute Rectangle Areas

For each subinterval, the area of the rectangle is f(xᵢ) · Δx, where f(xᵢ) is the function value (height) and Δx is the width. If f(xᵢ*) is negative, the rectangle contributes a negative area, correctly reflecting that the curve lies below the x-axis. This signed area convention is crucial for the definite integral's full meaning.

Sum All Rectangles

Add together all rectangle areas to get the Riemann sum: Sₙ = Σᵢ₌₁ⁿ f(xᵢ) Δx. This single number approximates the definite integral ∫ₐᵇ f(x) dx. As n → ∞ and Δx → 0, the approximation becomes exact (for integrable functions): ∫ₐᵇ f(x) dx = lim_{n→∞} Σᵢ₌₁ⁿ f(xᵢ) Δx.

Assess Accuracy

The error in a Riemann sum depends on n and the behavior of f. For a function with bounded derivative |f'(x)| ≤ M on [a, b], the error in left/right Riemann sums satisfies |Error| ≤ M(b−a)²/(2n). Doubling n roughly halves the error. The midpoint rule is generally more accurate, with error bounded by M₂(b−a)³/(24n²), where M₂ bounds |f''(x)|. This means increasing n improves accuracy rapidly for smooth functions.

The Key Equation

Sn=∑i=1nf(xi∗) Δx=∑i=1nf(xi∗)⋅b−anS_n = \sum_{i=1}^{n} f(x_i^*) \,\Delta x = \sum_{i=1}^{n} f(x_i^*) \cdot \frac{b-a}{n}
  • S_n — Riemann Sum (same as f(x)·x units): The total approximate area under the curve, computed using n rectangles.
  • n — Number of subintervals (dimensionless): The number of equal-width rectangles used in the approximation; larger n gives better accuracy.
  • x_i^* — Sample point (same as x units): A chosen point within the i-th subinterval [xᵢ₋₁, xᵢ]; could be left endpoint, right endpoint, or midpoint.
  • \Delta x — Subinterval width (same as x units): The equal width of each rectangle, calculated as (b − a)/n.
  • f(x_i^*) — Function value (height) (same as f(x) units): The height of the i-th rectangle, determined by evaluating the function at the sample point.

Worked Example

Problem. Approximate ∫₁³ x² dx using a Right Riemann Sum with n = 4 subintervals. Then compare your answer to the exact value.

S4=10.75,∫13x2 dx=263≈8.667S_4 = 10.75, \quad \int_1^3 x^2\,dx = \frac{26}{3} \approx 8.667

Answer. Right Riemann Sum with n = 4 gives S₄ = 10.75; the exact value is 26/3 ≈ 8.667, showing an overestimate of about 2.083.

Figure
Diagram
Right Riemann Sum for f(x) = x² on [1, 3], n = 4xy01.01.52.02.53.0149f(x)=x²2.254.006.259.00Each rectangle height = f(right endpoint); Δx = 0.5; shaded area ≈ 10.75
Origins · 1 minGeorg Friedrich Bernhard Riemann and how this was discovered
GF
Georg Friedrich Bernhard Riemann
1826–1866 · German

Riemann formalized the definition of the integral in his 1854 habilitation thesis 'On the Representability of a Function by a Trigonometric Series,' introducing what we now call the Riemann integral. His rigorous definition using partitions and sample points gave calculus a firm logical foundation, replacing earlier informal notions of area-under-a-curve that had been used since Newton and Leibniz. He also revolutionized geometry (Riemannian geometry) and made profound contributions to number theory through the Riemann Hypothesis. Riemann was so shy and anxious that he reportedly fell ill with nerves before his famous habilitation lecture in Göttingen—the very lecture where he introduced the ideas underlying Riemann sums. He died of tuberculosis at just 39, yet his work continues to influence mathematics nearly 170 years later.

  • The Definite Integral — The definite integral is the limit of Riemann sums as n → ∞; understanding Riemann sums is the direct path to defining and interpreting ∫ₐᵇ f(x) dx rigorously.
  • Fundamental Theorem of Calculus — The FTC connects antiderivatives to definite integrals, allowing exact evaluation of the limits that Riemann sums approximate, making it the key tool for moving beyond numerical approximation.
  • Trapezoidal Rule — The Trapezoidal Rule is a refined numerical integration method that averages left and right Riemann sums, achieving greater accuracy by using trapezoids instead of rectangles.
  • Sigma (Summation) Notation — Riemann sums are expressed compactly using sigma notation Σ; fluency with this notation is essential for writing and manipulating Riemann sum expressions.