The Riemann Integral: Rectangles Becoming an Area

The area under a curve, approximated by rectangles

The definite integral ∫f(x)dx from a to b is defined as the signed area between a curve and the x-axis. Bernhard Riemann's approach — the one taught first because it is the most direct — is to slice the interval [a,b] into n strips of width Δx = (b−a)/n, approximate the curve over each strip by a rectangle, and sum the rectangle areas:

Riemann sum = Σ f(xᵢ*) · Δx,   i = 1 to n

Δx = (b - a) / n
xᵢ* = a sample point inside the i-th subinterval — which one you pick defines the rule

As n grows and Δx shrinks toward zero, the sum converges to the exact integral for any function that is continuous (or has only finitely many jump discontinuities) on [a,b] — this convergence, made rigorous, is what actually defines the Riemann integral, before it is ever connected to antiderivatives via the fundamental theorem of calculus.

Left, right, midpoint: where you sample matters

The four common rules differ only in the choice of xᵢ* and, for the trapezoid rule, in what shape replaces the rectangle. The left rule samples the left edge of each strip, the right rule the right edge; for a monotonically increasing function these systematically under- and over-estimate the true area respectively (and vice-versa for a decreasing function), so the true integral is always trapped between them. The midpoint rule samples the centre of each strip and its errors from the two sides of the curve mostly cancel, making it noticeably more accurate for the same n.

The trapezoid rule does not use a rectangle at all — it connects f(xᵢ) and f(xᵢ₊₁) with a straight line and sums trapezoid areas instead:

trapezoid sum = Δx · [ f(x₀)/2 + f(x₁) + f(x₂) + ... + f(xₙ₋₁) + f(xₙ)/2 ]

Convergence rates: why midpoint and trapezoid beat left/right

Error analysis via Taylor expansion shows the left and right rules have error that shrinks as O(1/n) — halving the strip width only halves the error. The midpoint and trapezoid rules both have error O(1/n²) — halving the strip width quarters the error — because their sampling cancels the leading-order curvature term of the Taylor expansion that left/right sampling does not. Interestingly the midpoint rule's error is typically about half the size of the trapezoid rule's, and of opposite sign, which is exactly why Simpson's rule (not covered by these four, but a natural next step) combines them 2:1 to cancel the O(1/n²) term entirely and reach O(1/n⁴).

Where each rule struggles

Every rule assumes f is reasonably smooth over each strip. Functions like sqrt(x) near x = 0, where the derivative blows up, or |x| near a kink, degrade the convergence rate of every rule — the trapezoid and midpoint rules lose their O(1/n²) advantage and fall back toward O(1/n) near such points, because the Taylor expansion that guarantees the faster rate assumes bounded second derivatives. Oscillatory functions like sin(x) at high strip counts converge as expected once each strip is narrower than the function's own wiggle, but at low n a rectangle can straddle a peak and a trough and cancel out real area entirely.

Why this matters beyond a calculus exercise

Most integrals that show up in physics, engineering and statistics have no closed-form antiderivative — the Gaussian ∫e^(−x²)dx being the most famous example — so numerical quadrature is not a fallback, it is the only option. The same left/right/midpoint/trapezoid family, extended to two and three dimensions, underlies finite-element structural analysis, and the underlying idea of approximating an integral by a weighted sum of sampled function values scales all the way up to Monte Carlo integration for the high-dimensional integrals that appear in computational finance and rendering.

Frequently asked questions

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