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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 Engine tested · Specification checked · 13/13 tests · Production surface contract 4/4 · v1.1.0
Input interpretation
Enter values to calculate.
Result
—
Model
n-point Gauss–Legendre quadrature 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.

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 1.1.0 · 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].
Inputs
  • integrand
  • a
  • b
  • n?
  • coeffs?
Outputs
  • value
  • exact
  • abs_err
Formula
Gauss–Legendre n-point
Assumptions
  • 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 4/4 · 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
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, Golden Section Search Calculator, Numerical Derivative Calculator, Numerical Interpolation Calculator, Numerical Root Calculator, One-Dimensional Optimization Calculator, Simpson Rule 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. › → 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. › → 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. ›
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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.