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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.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · production STALE · 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
Definite integral 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; trapezoid/Simpson panel overlays when n≤16; Gauss nodes when method=gauss). Same fnValue as REST/MCP. Not indefinite CAS.
Verification
Engine tested · Source checked · v1.1.1 · CVP VERIFIED · production STALE · 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 STALE · Overall CVP state: VERIFIED_STALE under CVP 1.0.0-proposed
CVP status
STALE · Capability production binding: STALE · Site report: STALE · Evidence changed after the last successful production attestation. Re-attestation required.
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
  • ISO 80000-2:2019 Quantities and units — Mathematics
  • CRC Standard Mathematical Tables — composite trapezoid, Simpson 1/3, Gauss–Legendre
Sources
Evidence
5 legacy golden · 2 legacy boundary · legacy regression suite · 83/83 oracle-backed golden · 5/5 invalid · Artifact integrity PASS
STALE · Build schema 1.1.1 matches page · production binding STALE · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · CVP STALE · attestation STALE — Production CURRENT withheld · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

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

Formulas

Core equations used by this calculator.

Trapezoidh/2 · (y₀ + 2Σ yᵢ + yₙ)
Simpson 1/3h/3 · (y₀ + 4Σ odd + 2Σ even + yₙ)
Gauss–Legendre((b−a)/2) Σ wᵢ f(xᵢ)
iNot indefinite integration and not symbolic simplify. Polynomial coefficients are constant-first. Simpson/trapezoid n is panels (2–10000). Gauss n is nodes (2–8).

How to use

1

Choose Simpson, trapezoid, or Gauss

Poly needs coeffs. sin and exp are f(x)=sin x and e^x. Discovery URLs lock one method.

2

Enter [a, b] and n

Simpson n is forced even. Gauss n is 2–8 nodes. The exact antiderivative is shown beside the approximation.

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

Simpson

−cos π + cos 0
2

Definite Integral calculator specification

Version 1.1.1 · Engine tested

Calculation status

Review policy · Evidence

Definition
The definite integral ∫_a^b f(x) dx is approximated by composite trapezoid, Simpson 1/3, or n-point Gauss–Legendre. Polynomial, sin, and exp have an exact antiderivative used as a check — not a CAS.
What it calculates
Definite integral of poly, sin, or exp on [a, b], with an engine-linked shaded figure of ∫_a^b f.
Inputs
  • method
  • integrand
  • a
  • b
  • n?
  • coeffs?
Outputs
  • value
  • exact
  • abs_err
Formula
composite trapezoid / Simpson 1/3 / Gauss–Legendre
Assumptions
  • The Result Card includes an engine-linked figure of the integrand on [a, b] (true f shaded; trapezoid/Simpson panel overlays when n≤16; Gauss nodes when method=gauss). Same fnValue as REST/MCP. Not indefinite CAS.
  • Polynomial degree ≤ 8
  • exact is a binary64 closed-form baseline, not symbolic exact arithmetic. abs_err retains the historical signed value: raw quadrature minus baseline; it is not an error bound.
  • Reversed intervals change the sign; equal bounds return zero after input validation.
  • Gauss n is nodes 2–8, not panels
Units
  • dimensionless
Boundary conditions
  • Simpson/trapezoid n not in 2–10000 → INVALID_INPUT
  • Gauss n > 8 → VALUE_ABOVE_MAX
  • Nonfinite evaluation or baseline → NUMERICAL_OVERFLOW
  • unknown method (including romberg) → INVALID_MODE
Example
simpson poly 0,1 on [0,1] → 0.5
Validation cases

3 published on this page · 13/13 tests · Production surface contract 6/6 · View evidence

  • method=simpson integrand=poly coeffs=0,1 a=0 b=1 n=10 → exact=0.5
  • method=gauss integrand=poly coeffs=0,1 a=0 b=1 n=2 → value=0.5
  • method=romberg integrand=sin a=0 b=1 → INVALID_MODE
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Definite integral ∫_a^b f(x) dx
  • CRC Standard Mathematical Tables — composite trapezoid, Simpson 1/3, Gauss–Legendre
    Supports: Composite trapezoid and Simpson 1/3; n-point Gauss–Legendre exact for degree ≤ 2n−1
Calculation version
1.1.1

Other calculators in this family: Brent Method Calculator, Euler Method Calculator, Gauss Quadrature 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. › → 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. › → 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.

Can this do indefinite integrals?

No. Symbolic antiderivatives are out of scope. This page is a definite numerical integral with an exact check for poly, sin, and exp.

Where does this run?

Locally in the browser by default. REST and MCP call the same definite-integral engine. /simpson-rule, /trapezoidal-rule, and /gauss-quadrature are discovery URLs, not second engines.