Trapezoidal Rule Calculator – Numerical Integration

Trapezoidal rule calculator for numerical integration. Approximate definite integrals from a function f(x) or from data points, see step size, composite formula, grid table, and interpretation.

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Trapezoidal Rule Calculator – Numerical Integration

Approximate definite integrals using the trapezoidal rule. Work from a function \(f(x)\) on an equally spaced grid or from tabulated data points \((x_i, f(x_i))\).

Designed for calculus students, instructors, and engineers who need a transparent, step-by-step numerical integration tool.

Trapezoidal rule calculator

Composite & data-based trapezoidal rule

Choose the calculation mode, fill in the inputs, and click Calculate. The tool reports the trapezoidal approximation, effective step size, and a detailed grid table.

Calculation mode

n ≥ 1. A larger n gives a finer grid and usually better accuracy.

Use standard notation. The calculator automatically maps sin, cos, tan, exp, ln, log, sqrt, abs, pi, e to JavaScript’s Math.* functions and interprets ^ as a power operator.

Numerical tolerance for diagnostics: 1e-12

The trapezoidal approximation and grid table will appear here after you click Calculate.

Understanding the trapezoidal rule

The trapezoidal rule is one of the simplest and most widely used numerical integration formulas. Instead of computing the exact area under the curve \(y = f(x)\), you approximate the graph with straight-line segments and sum the areas of the resulting trapezoids.

1. Basic trapezoid formula

On a single interval \([x_i, x_{i+1}]\) with function values \(f(x_i)\) and \(f(x_{i+1})\), the area of the trapezoid is:

\[ \int_{x_i}^{x_{i+1}} f(x)\,dx \approx \frac{x_{i+1} - x_i}{2}\,\bigl(f(x_i) + f(x_{i+1})\bigr). \]

This formula works whether your data come from a function or from measurements. The only requirement is that you know the x-coordinates and the corresponding function (or measured) values.

2. Composite trapezoidal rule on an equally spaced grid

If you divide \([a, b]\) into \(n\) equal subintervals of width \(h = \frac{b-a}{n}\), with grid points \(x_0 = a, x_1, \dots, x_n = b\), the composite trapezoidal rule is:

\[ \int_a^b f(x)\,dx \approx \frac{h}{2} \left[f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n)\right]. \]

The function-based mode of this calculator builds exactly this grid, evaluates \(f(x_i)\) at each point, and applies the formula.

3. Error behavior

If \(f\) has a continuous second derivative on \([a, b]\), the global error of the composite trapezoidal rule satisfies

\[ \text{Error} \approx -\frac{(b - a)}{12} h^2\, f''(\xi) \quad \text{for some} \;\xi \in (a, b). \]

This shows a quadratic dependence on the step size \(h\): halving \(h\) (doubling the number of subintervals) typically reduces the error by roughly a factor of 4, provided \(f\) is smooth enough and does not oscillate too wildly.

4. Trapezoidal rule vs. Simpson’s rule

Simpson’s rule uses quadratic interpolation on pairs of subintervals and generally achieves higher accuracy (error proportional to \(h^4\)) for smooth integrands. The trapezoidal rule, however, has important advantages:

  • It works directly with tabulated data and nonuniform x-spacing.
  • It is easy to implement and explain in a classroom setting.
  • It is often sufficient for engineering estimates and quick checks.

In practice, you can use the trapezoidal rule to get a first estimate, then refine or cross-check with Simpson’s rule when it is available.

Trapezoidal rule – FAQ

Audit: Complete
Formula (LaTeX) + variables + units
This section shows the formulas used by the calculator engine, plus variable definitions and units.
Formula (extracted LaTeX)
\[\int_{x_i}^{x_{i+1}} f(x)\,dx \approx \frac{x_{i+1} - x_i}{2}\,\bigl(f(x_i) + f(x_{i+1})\bigr).\]
\int_{x_i}^{x_{i+1}} f(x)\,dx \approx \frac{x_{i+1} - x_i}{2}\,\bigl(f(x_i) + f(x_{i+1})\bigr).
Formula (extracted LaTeX)
\[\int_a^b f(x)\,dx \approx \frac{h}{2} \left[f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n)\right].\]
\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].
Formula (extracted LaTeX)
\[\text{Error} \approx -\frac{(b - a)}{12} h^2\, f''(\xi) \quad \text{for some} \;\xi \in (a, b).\]
\text{Error} \approx -\frac{(b - a)}{12} h^2\, f''(\xi) \quad \text{for some} \;\xi \in (a, b).
Formula (extracted LaTeX)
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Formula (extracted text)
\[ \int_{x_i}^{x_{i+1}} f(x)\,dx \approx \frac{x_{i+1} - x_i}{2}\,\bigl(f(x_i) + f(x_{i+1})\bigr). \]
Formula (extracted text)
\[ \int_a^b f(x)\,dx \approx \frac{h}{2} \left[f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n)\right]. \]
Formula (extracted text)
\[ \text{Error} \approx -\frac{(b - a)}{12} h^2\, f''(\xi) \quad \text{for some} \;\xi \in (a, b). \]
Variables and units
  • T = property tax (annual or monthly depending on input) (currency)
Sources (authoritative):
Changelog
Version: 0.1.0-draft
Last code update: 2026-01-19
0.1.0-draft · 2026-01-19
  • Initial audit spec draft generated from HTML extraction (review required).
  • Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
  • Confirm sources are authoritative and relevant to the calculator methodology.
Verified by Ugo Candido on 2026-01-19
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Formulas

(Formulas preserved from original page content, if present.)

Version 0.1.0-draft
Citations

Add authoritative sources relevant to this calculator (standards bodies, manuals, official docs).

Changelog
  • 0.1.0-draft — 2026-01-19: Initial draft (review required).