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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.

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
n-point Gauss–Legendre quadrature 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; tabulated Gauss–Legendre nodes). Same engine as /definite-integral. n is nodes 2–8, not panels. 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.

Mapx = ((b−a)/2) ξ + (a+b)/2
Sum((b−a)/2) Σ wᵢ f(xᵢ)
in is the node count 2–8, not Simpson/trapezoid panels. Polynomial coefficients are constant-first. Not a CAS. Not Romberg.

How to use

1

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

Poly needs coeffs. Default n=5. Simpson and trapezoid live on sibling discovery URLs.

2

Read the approximation and exact check

A 2-point rule is already exact for cubics. Poly, sin, and exp show an exact antiderivative beside the Gauss value.

Example calculations

Common configurations with formula and result.

ϟ

∫₀¹ x dx

2-point is exact

1/2
0.5
ϟ

∫₀¹ x² dx

2-point is exact for degree 2

1/3
0.333…

Gauss Quadrature calculator specification

Version 1.1.1 · Engine tested

Calculation status

Review policy · Evidence

Definition
n-point Gauss–Legendre quadrature approximates ∫_a^b f(x) dx from nodes on [-1, 1] mapped to [a, b]. Exact for polynomials of degree ≤ 2n−1. This page is method=gauss on math.numerical.definite_integral. Not a second integral engine. Not composite panels.
What it calculates
n-point Gauss–Legendre quadrature 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
Gauss–Legendre n-point
Assumptions
  • The Result Card includes an engine-linked figure of the integrand on [a, b] (true f shaded; tabulated Gauss–Legendre nodes). Same engine as /definite-integral. n is nodes 2–8, not panels. Not indefinite CAS.
  • n is nodes 2–8, not panels
  • Polynomial degree ≤ 8
Units
  • dimensionless
Boundary conditions
  • n < 2 → INVALID_INPUT
  • n > 8 → VALUE_ABOVE_MAX
  • unknown method (including romberg) → INVALID_MODE
Example
poly 0,1 on [0,1] n=2 → 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=2 → exact=0.5 value=0.5
  • integrand=sin a=0 b=1 n=9 → VALUE_ABOVE_MAX
Sources
Calculation version
1.1.1

Background

Interpretation and common distinctions.

What does this gauss quadrature calculator compute?

n-point Gauss–Legendre quadrature approximates ∫_a^b f(x) dx from nodes on [-1, 1] mapped to [a, b]. Exact for polynomials of degree ≤ 2n−1.

The published formula or method is Gauss–Legendre n-point. 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 2-point is exact 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, Golden Section Search Calculator, Nonlinear Equation Calculator, Numerical Derivative Calculator, Numerical Interpolation Calculator, Numerical Root 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. › → 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. › → Numerical Root Calculator Find an approximate real zero of a polynomial or sin x with Newton, bisection, or secant. Enter a starting point or bracket and inspect the stopping criterion. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second integral engine?

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

Why is n only 2–8?

This seed uses tabulated Gauss–Legendre nodes. Composite trapezoid/Simpson still use panels 2–10000 on the canonical page.