Why does the number of subintervals n need to be an even number?

Composite Simpson's 1/3 rule fits parabolas across pairs of adjacent subintervals, which requires an even number of divisions.

What functions and operators can I use in f(x)?

You can use standard mathematical operations and built-in functions such as sin, cos, exp, ln, sqrt, and power expressions.

Why is the estimate exact for quadratic and cubic polynomials?

Simpson's 1/3 rule has a degree of precision of 3, meaning its truncation error term relies on the fourth derivative, which is zero for polynomials up to degree 3.

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

Yes. When b is less than a, the step size h is negative, and the integral flips sign according to standard calculus rules.

What is the purpose of the exact antiderivative field?

Entering F(x) computes the theoretical value F(b) - F(a) to display the exact absolute error alongside the numerical approximation.

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Simpson's Rule Integrator (∫f(x)dx)

Composite Simpson's 1/3 rule for ∫f(x)dx (even n): node table with weights, integral estimate, optional exact value, O(h⁴) accuracy.

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2 examples

∫₀¹ x² dx, n = 4 → S = 0.333333, exact for quadratics

Simpson's rule integrates any quadratic (in fact any cubic) exactly: with h = 0.25 the 1/3-rule sum lands on 1/3 to full precision, zero error against the exact antiderivative x³/3.

Composite Simpson's 1/3 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) weight 0 0.000000 0.000000 1 1 0.250000 0.062500 4 2 0.500000 0.250000 2 3 0.750000 0.562500 4 4 1.000000 1.000000 1 Simpson estimate: S ≈ 0.333333 Exact F(b) - F(a) = 0.333333, |error| ≈ 0.000000

∫₀¹ e^(−x²) dx, n = 10 → S ≈ 0.746825 (no elementary antiderivative)

The Gaussian integral has no elementary antiderivative, so quadrature is the way: with n = 10 Simpson gives 0.746825, within about 10⁻⁶ of the exact √π/2·erf(1) = 0.7468241 — O(h⁴) accuracy with a coarse grid.

Composite Simpson's 1/3 rule ∫ from 0 to 1 of f(x) dx, f(x) = e^(-x^2), n = 10 subintervals, h = 0.1 Simpson estimate: S ≈ 0.746825 (No elementary antiderivative exists; √π/2·erf(1) ≈ 0.7468241)

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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 n (even)numberRequiredMust be even. h = (b − a)/n; global error is O(h⁴) — halving h divides the error by ~16.Decimal PlacesnumberOptional

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Simpson's Rule Integrator (∫f(x)dx)

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Examples

1. Exact Integration of a Quadratic Polynomial

2. Approximating a Non-Elementary Gaussian Integral

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