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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 CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published methods
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published methods
Assurance
Core
Declared partition coverage
PASS · 4/4 declared partitions (simpson, trapezoid, gauss, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Composite Simpson 1/3 of poly, sin, or exp on [a, b], with an engine-linked shaded figure of ∫_a^b f.
Scope
The Result Card includes an engine-linked figure of the integrand on [a, b] (true f shaded; Simpson interpolant overlay when n≤16). Same engine as /definite-integral. Not indefinite CAS.
Verification
Engine tested · Source checked · v1.1.1 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published methods· View Manifest · CVP overview · Specification
Versions
Calculation 1.1.1 · CVP protocol 1.0.0-proposed · Evidence 2026-10-07.integral-stability-iface2
Verification revision
2026-10-07.integral-stability-iface2 · 1/1 property · digest f22f649f5f61
Legacy regression
13/13 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 2/2 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
83/83 golden · 2/2 CVP boundary · 5/5 invalid · 1/1 property · 3/3 metamorphic · 7/7 cross-interface · 1/1 cross-calculator · 6/6 CVP contract · Manifest
Sources
Sources
Evidence
5 legacy golden · 2 legacy boundary · legacy regression suite · 83/83 oracle-backed golden · 5/5 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.1.1 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

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 1.1.1 · 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], with an engine-linked shaded figure of ∫_a^b f.
Inputs
  • integrand
  • a
  • b
  • n?
  • coeffs?
Outputs
  • value
  • exact
  • abs_err
Formula
Simpson 1/3
Assumptions
  • The Result Card includes an engine-linked figure of the integrand on [a, b] (true f shaded; Simpson interpolant overlay when n≤16). Same engine as /definite-integral. 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 6/6 · 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
Sources
Calculation version
1.1.1

Background

Interpretation and common distinctions.

What does this simpson rule calculator compute?

Composite Simpson 1/3 approximates ∫_a^b f(x) dx with even panel count n.

The published formula or method is Simpson 1/3. The following references support the stated mathematics or method:

Which inputs and limits should you check?

Choose a method suitable for the integrand and interval. Refinement diagnostics are not a universal error bound for discontinuous or otherwise unsupported functions.

The worked example uses coeffs 0,1 with 1/2. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.numerical.definite_integral v1.1.1 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Other calculators in this family: Brent Method Calculator, Definite Integral Calculator, Euler Method Calculator, Gauss Quadrature Calculator, Golden Section Search Calculator, Nonlinear Equation Calculator, Numerical Derivative Calculator, Numerical Interpolation 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. › → Euler Method Calculator Forward Euler for a scalar first-order IVP y′=f(t,y). Same engine as Ordinary Differential Equation. Not RK4, not rk45, not a system. 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. › → Nonlinear Equation Calculator Find one approximate real zero of a polynomial or sin x with Newton, bisection, or secant. Uses the Numerical Root engine. Runs locally. › → Numerical Derivative Calculator Finite-difference derivative of a polynomial, sin, or exp, plus a numeric two-sided limit. Central, forward, backward, or limit. Discovery /two-sided-limit. 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. ›

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.