LaTeX
\int_a^b f(x)dx\approx\frac{h}{2}\left[f(x_0)+2\sum_{i=1}^{n-1}f(x_i)+f(x_n)\right]Variables
- h=(b−a)/n: step width
- x_i: grid point
- n: number of subintervals
How to use this formula
Approximates a definite integral with piecewise linear segments.
Important notes
- The composite rule assumes equal spacing.
- Error decreases quadratically for sufficiently smooth functions.
Quick example
With one interval, the estimate is the trapezoid area (b−a)(f(a)+f(b))/2.
Applicability, worked calculation, and verification
Assumptions and domain checks
- The composite rule assumes equal spacing.
- For the Trapezoidal Rule, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- For the Trapezoidal Rule, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.
- Integration limits, the differential, and any constant of integration must be included where required.
Worked example
With one interval, the estimate is the trapezoid area (b−a)(f(a)+f(b))/2.
Common mistakes
- When copying Trapezoidal Rule, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- Do not report a numerical approximation without a tolerance, convergence check, and an estimate of rounding or truncation error.
Continue the workflow
Use Trapezoidal Rule in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\int_a^b f(x)dx\approx\frac{h}{2}\left[f(x_0)+2\sum_{i=1}^{n-1}f(x_i)+f(x_n)\right]. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Trapezoidal Rule used for?
Approximates a definite integral with piecewise linear segments.
Can I copy this formula as LaTeX?
Yes. Copy \int_a^b f(x)dx\approx\frac{h}{2}\left[f(x_0)+2\sum_{i=1}^{n-1}f(x_i)+f(x_n)\right] or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.