Riemann Sums and Integrability Criteria: Full Guide

A Riemann sum approximates ∫ab f by sampling the function: for a partition into subintervals of widths Δxi, pick any sample point ξi in each subinterval and add up f(ξi)Δxi. Left sums, right sums and midpoint sums are just special choices of sample points. The integrability criteria say exactly when these sums settle down to a single number as the partition gets finer.

This viewpoint is the bridge between theory and computation. Every numerical integration method is a clever Riemann sum, and the criteria tell us which functions those methods can handle — a question that matters every time software integrates something it cannot antidifferentiate.

Key Concepts

• Riemann sum: Σ f(ξi)Δxi with ξi ∈ [xi−1, xi]. Different choices of ξi give left, right, midpoint or trapezoidal rules.
• Mesh (norm) of a partition: the largest subinterval width. Integrability means all Riemann sums converge to the same limit as the mesh → 0, regardless of sample points.
• Lebesgue's criterion: a bounded function on [a, b] is Riemann integrable if and only if it is continuous almost everywhere — its discontinuities form a set of measure zero.
• Dirichlet function: 1 on rationals, 0 on irrationals; discontinuous everywhere, so not Riemann integrable on any interval.
• Thomae's function: 1/q at p/q in lowest terms, 0 at irrationals; discontinuous exactly at the rationals (measure zero), hence integrable with integral 0.

Formulas

• Right Riemann sum (regular partition): Rn = ((b−a)/n) Σi=1n f(a + i(b−a)/n).
• Trapezoid rule: Tn = (Δx/2)(f(x₀) + 2f(x₁) + … + 2f(xn−1) + f(xn)), the average of left and right sums.
• Midpoint rule: Mn = Δx Σ f((xi−1 + xi)/2), usually the most accurate simple choice.
• Lebesgue criterion: bounded f on [a, b] is Riemann integrable ⟺ the set of discontinuities has measure zero.

Worked Example

Approximate ∫02 x² dx with the right Riemann sum using n = 4, and compare with the exact value.

1. The subinterval width is Δx = (2 − 0)/4 = 0.5, with right endpoints 0.5, 1.0, 1.5, 2.0.
2. Function values: f(0.5) = 0.25, f(1) = 1, f(1.5) = 2.25, f(2) = 4.
3. Right sum: R4 = 0.5 × (0.25 + 1 + 2.25 + 4) = 0.5 × 7.5 = 3.75.
4. The exact integral is [x³/3]02 = 8/3 ≈ 2.6667. The right sum overestimates because x² is increasing; refining to n = 40 gives ≈ 2.717, visibly closing in on 8/3.

Common Misconceptions

• "A finer partition always gives a better approximation." Only for integrable functions; for the Dirichlet function every Riemann sum is either 0 or 1 depending on sample points, no matter how fine the partition.
• "Left and right sums are equally good." For monotone functions one always overestimates and the other underestimates; the midpoint rule is typically far more accurate.
• "Riemann sums require a formula for f." They need only sampled values, which is why they work on measured data where no formula exists.

Real-World Applications

• Numerical integration in scientific software for functions with no elementary antiderivative, such as e−x² in statistics.
• Estimating total distance from discrete velocity readings, or total energy from sampled power data.
• Computer graphics: approximating integrals for lighting and shading by sampling, a direct descendant of Riemann sums.

Riemann sums turn integration into sampling: pick points, multiply heights by widths, add up. They converge to the integral exactly for the functions Lebesgue's criterion allows — the bounded ones whose discontinuities are negligible. The worked example shows the typical behaviour: a coarse sum is rough, refinement improves it, and for non-integrable functions like Dirichlet's, no amount of refinement helps.