What is the global truncation error of the trapezoidal rule?

The composite trapezoidal rule has an error order of O(h²), meaning doubling the number of subintervals roughly reduces the error by a factor of four.

Can the upper limit b be smaller than the lower limit a?

Yes. When b < a, the step size h becomes negative, correctly reversing the sign of the definite integral.

What math functions are supported in f(x)?

Standard expressions including polynomials, powers (^), sin, cos, exp, ln, sqrt, and basic arithmetic operations are supported.

Is entering an antiderivative F(x) mandatory?

No. The antiderivative field is optional and only used to compute the analytical benchmark and absolute error.

How many subintervals can I specify?

You can set n from 1 up to 100,000 subintervals to balance computational precision and table detail.

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Trapezoidal Rule Integrator (∫f(x)dx)

Composite trapezoidal rule for ∫f(x)dx: node table, integral estimate, optional exact value and O(h²) error report.

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∫₀¹ x² dx, n = 4 → T = 0.343750 (exact 1/3, error 0.010417)

With h = 0.25 the trapezoid sum is T = 0.25/2·(0 + 2·(0.0625 + 0.25 + 0.5625) + 1) = 0.34375. The exact value is 1/3 = 0.333333, so the absolute error is 0.010417 — the expected O(h²) behavior.

Composite trapezoidal rule ∫ from 0 to 1 of f(x) dx, f(x) = x^2, n = 4 subintervals, h = 0.25 i x_i f(x_i) 0 0.000000 0.000000 1 0.250000 0.062500 … 4 1.000000 1.000000 Trapezoid estimate: T ≈ 0.343750 Exact F(b) - F(a) = 0.333333, |error| ≈ 0.010417

∫₀¹ e^x dx, n = 4 → T ≈ 1.727222 (exact e − 1)

With h = 0.25 the trapezoid rule gives 1.727222 for ∫e^x dx on [0, 1]; the exact value is e − 1 = 1.718282, so the error is about 0.008940. Doubling n roughly quarters the error — the signature of a second-order method.

Composite trapezoidal rule ∫ from 0 to 1 of f(x) dx, f(x) = e^x, n = 4 subintervals, h = 0.25 Trapezoid estimate: T ≈ 1.727222 Exact F(b) - F(a) = 1.718282, |error| ≈ 0.008940

Inputs

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6 options

ContentPaste or type the main input values.2

f(x) to integratetextRequiredIntegrand in x (functions: sin, cos, exp, ln, sqrt, …).Exact Antiderivative F(x) (optional)textOptionalExact antiderivative in x; the tool reports F(b) − F(a) and the absolute error. Leave empty to skip.

SettingsAdjust formats, ranges, numbers, and modes.4

Lower Limit anumberRequiredUpper Limit bnumberRequiredMay be less than a (the integral flips sign, as in calculus).Subintervals nnumberRequiredh = (b − a)/n. More subintervals → smaller error, O(h²) globally.Decimal PlacesnumberOptional

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Trapezoidal Rule Integrator (∫f(x)dx)

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Examples

1. Integrating a Quadratic Polynomial

2. Integrating an Exponential Function

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