Trapezium Rule

Many real-world integrals cannot be evaluated using standard antiderivative techniques — the function may be too complex, defined only by data points, or simply have no closed-form antiderivative. Numerical integration provides a family of methods for estimating the value of a definite integral using arithmetic operations on sampled function values. The Trapezium Rule (also called the Trapezoidal Rule) is one of the most fundamental and widely used of these techniques. The central idea behind the Trapezium Rule is elegant: instead of computing the exact area under a curve, we approximate that area by dividing it into vertical strips and capping each strip with a straight line between two known points on the curve. Each strip then forms a trapezium (a quadrilateral with one pair of parallel sides), and the area of a trapezium is easy to calculate. Summing the areas of all the strips gives an estimate for the definite integral. The accuracy of the Trapezium Rule depends on the number of strips used and the curvature of the function. For a function that is nearly linear over the interval, even a small number of strips gives an excellent approximation. For highly curved functions, more strips are needed. Understanding the rule — its derivation, application, and limitations — is an essential step in the study of numerical methods in calculus.

Key takeaways
  • Setting Up the Interval
  • Area of One Trapezium
  • Summing All Strips
  • The Trapezium Rule Formula
  • Error and Improvement

Setting Up the Interval

To approximate ∫[a to b] f(x) dx, divide the interval [a, b] into n equal subintervals. Each subinterval has width h = (b − a)/n. Label the x-values x₀ = a, x₁ = a + h, x₂ = a + 2h, …, xₙ = b. Compute the corresponding y-values: y₀ = f(x₀), y₁ = f(x₁), …, yₙ = f(xₙ). These n+1 points are called ordinates.

Area of One Trapezium

On each subinterval [xᵢ, xᵢ₊₁], the curve is approximated by the straight line segment connecting (xᵢ, yᵢ) to (xᵢ₊₁, yᵢ₊₁). This forms a trapezium with parallel sides of lengths yᵢ and yᵢ₊₁, and width h. The area of this single trapezium is: A_i = ½ × h × (yᵢ + yᵢ₊₁).

Summing All Strips

Sum the areas of all n trapeziums: Total ≈ ½h(y₀ + y₁) + ½h(y₁ + y₂) + … + ½h(yₙ₋₁ + yₙ). Notice that y₁, y₂, …, yₙ₋₁ each appear twice (as the right edge of one trapezium and the left edge of the next), while y₀ and yₙ appear only once. Factoring gives the compact Trapezium Rule formula.

The Trapezium Rule Formula

The standard form is: ∫[a to b] f(x) dx ≈ (h/2)[y₀ + 2y₁ + 2y₂ + … + 2yₙ₋₁ + yₙ], where h = (b − a)/n. This can be written as (h/2)[y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)]. A memory aid: the first and last ordinates are added once; all middle ordinates are doubled.

Error and Improvement

The Trapezium Rule has a truncation error proportional to h². This means doubling the number of strips (halving h) roughly quarters the error. If the function is concave up (f''(x) > 0), the trapeziums lie above the curve and the rule overestimates; if the function is concave down (f''(x) < 0), the rule underestimates. The error bound is: |Error| ≤ (b−a)³/(12n²) × max|f''(x)| on [a,b]. Increasing n always improves accuracy.

The Key Equation

∫abf(x) dx≈h2[y0+2y1+2y2+⋯+2yn−1+yn],h=b−an\int_{a}^{b} f(x)\,dx \approx \frac{h}{2}\bigl[y_0 + 2y_1 + 2y_2 + \cdots + 2y_{n-1} + y_n\bigr], \quad h = \frac{b-a}{n}
  • h — Strip width (same as x): The common width of each subinterval, equal to (b − a) divided by n.
  • n — Number of strips (dimensionless): The number of equal subintervals the interval [a, b] is divided into. More strips generally means better accuracy.
  • y_i — Ordinate (same as f(x)): The value of f at the i-th x-value, i.e. yᵢ = f(a + ih). There are n+1 ordinates in total.
  • y_0,\, y_n — End ordinates (same as f(x)): The function values at x = a and x = b respectively. These appear with coefficient 1 in the formula.

Worked Example

Problem. Use the Trapezium Rule with n = 4 strips to estimate ∫[1 to 3] (1/x) dx. Give your answer to 4 decimal places. (The exact value is ln 3 ≈ 1.0986.)

∫131x dx≈1.1167\int_{1}^{3} \frac{1}{x}\,dx \approx 1.1167

Answer. Approximately 1.1167 (overestimate; exact value ln 3 ≈ 1.0986)

Figure
Diagram
Trapezium Rule: Approximating Area Under y = 1/xxy11.522.5310.5y₀y₁y₂y₃y₄Trapezium approximationTrue curve y = 1/xn = 4 strips, h = 0.5, interval [1, 3]
Origins · 1 minRoger Cotes and how this was discovered
RC
Roger Cotes
1682–1716 · British (English)

Roger Cotes developed and refined many numerical integration methods in the early 18th century, including the family of Newton–Cotes formulas of which the Trapezium Rule is the simplest case. Working closely with Isaac Newton, he edited the second edition of the Principia Mathematica and made substantial original contributions to mathematics, including early work on logarithms and trigonometric identities. Isaac Newton reportedly said of Cotes: 'If he had lived, we might have known something.' Cotes died at only 33 years old, leaving behind a remarkable but tragically short mathematical legacy.

  • Riemann Sums — Riemann sums use rectangles rather than trapeziums to approximate definite integrals; understanding both methods reveals how the shape of the approximating region affects accuracy.
  • Simpson's Rule — Simpson's Rule is a more accurate numerical integration method that fits parabolic arcs instead of straight lines across pairs of strips, giving an error proportional to h⁴ rather than h².
  • Definite Integrals — The Trapezium Rule estimates the value of a definite integral, so a firm understanding of what a definite integral represents as a signed area is essential context.
  • Concavity and the Second Derivative — Whether the Trapezium Rule over- or underestimates depends directly on the concavity of the function, which is determined by the sign of its second derivative.