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Simpson Rule Calculator

Composite Simpson 1/3 of a polynomial, sin, or exp on [a, b]. Same engine as Definite Integral. Not indefinite CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 13/13 tests · Production surface contract 4/4 · v1.1.0
Input interpretation
Enter values to calculate.
Result
—
Model
Composite Simpson 1/3 of poly, sin, or exp on [a, b].
Scope
Not indefinite CAS
Verification
Engine tested · 13/13 tests · Production surface contract 4/4 · Specification checked · v1.1.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
5 golden · 2 boundary · 6 property · Production surface contract 4/4 · Artifact integrity PASS
Production
Embedded snapshot: STALE · Last attested schema matched 1.1.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status STALE (1 capability; 163 remain CURRENT) @ 2026-09-19T00:00:17.039Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

Simpson 1/3h/3 · (y₀ + 4Σ odd + 2Σ even + yₙ)
in is the panel count and is forced even (2–10000). Polynomial coefficients are constant-first. Not a CAS. Not Romberg.

How to use

1

Enter integrand, [a, b], and panels n

Poly needs coeffs. sin and exp are f(x)=sin x and eˣ. Larger method choice lives at /calc/math/definite-integral.

2

Read the approximation and exact check

Poly, sin, and exp show an exact antiderivative beside the Simpson value.

Example calculations

Common configurations with formula and result.

ϟ

∫₀¹ x dx

coeffs 0,1

1/2
0.5
ϟ

∫₀^π sin x dx

n=200

−cos π + cos 0
2

Simpson Rule calculator specification

Version 1.1.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Composite Simpson 1/3 approximates ∫_a^b f(x) dx with even panel count n. This page is method=simpson on math.numerical.definite_integral. Not a second integral engine. Not indefinite symbolic integrate.
What it calculates
Composite Simpson 1/3 of poly, sin, or exp on [a, b].
Inputs
  • integrand
  • a
  • b
  • n?
  • coeffs?
Outputs
  • value
  • exact
  • abs_err
Formula
Simpson 1/3
Assumptions
  • Not indefinite CAS
  • Polynomial degree ≤ 8
  • Not a second integral engine
Units
  • dimensionless
Boundary conditions
  • n not in 2–10000 → INVALID_INPUT
  • unknown method (including romberg) → INVALID_MODE
Example
poly 0,1 on [0,1] n=10 → 0.5
Validation cases

2 published on this page · 13/13 tests · Production surface contract 4/4 · View evidence

  • integrand=poly coeffs=0,1 a=0 b=1 n=10 → exact=0.5
  • method=romberg integrand=sin a=0 b=1 → INVALID_MODE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.1.0

Other calculators in this family: Brent Method Calculator, Definite Integral Calculator, Gauss Quadrature Calculator, Golden Section Search Calculator, Numerical Derivative Calculator, Numerical Interpolation Calculator, Numerical Root Calculator, One-Dimensional Optimization Calculator . Explore all Numerical Calculus.

→ Brent Method Calculator Brent’s method (golden section plus inverse parabolic interpolation) for a minimum of a polynomial, sin, or exp on [a, b]. Same engine as One-Dimensional Optimization. Not root-finding Brent. Runs locally. › → Definite Integral Calculator Composite trapezoid, Simpson 1/3, or Gauss–Legendre of a polynomial, sin, or exp on [a, b]. Not indefinite symbolic integrate. Runs locally. › → Gauss Quadrature Calculator n-point Gauss–Legendre quadrature of a polynomial, sin, or exp on [a, b]. Same engine as Definite Integral. Not indefinite CAS. Runs locally. › → Golden Section Search Calculator Golden-section search for a minimum of a polynomial, sin, or exp on [a, b]. Same engine as One-Dimensional Optimization. Not Brent, not multivariate. Runs locally. › → Numerical Derivative Calculator Finite-difference derivative of a polynomial, sin, or exp. Central, forward, or backward. Not symbolic d/dx. Runs locally. › → Numerical Interpolation Calculator Piecewise-linear, Lagrange, or natural cubic spline of tabulated (x, y). Not regression, not CAS. Runs locally. › → Numerical Root Calculator Find a real root of a polynomial with Newton, bisection, or secant. One capability, three methods. Not the nth-root page. Runs locally. › → One-Dimensional Optimization Calculator Minimize a polynomial, sin, or exp on [a, b] by golden-section search or Brent. Not multivariate, not max, not CAS. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second integral engine?

No. It is method=simpson on math.numerical.definite_integral, the same engine as /calc/math/definite-integral.

Can this do indefinite integrals?

No. Symbolic antiderivatives are out of scope.