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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 · 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 · v1.1.5 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published methods· View Manifest · CVP overview · Specification
Versions
Calculation 1.1.5 · CVP protocol 1.0.0-proposed · Evidence 2026-10-28.metric-scope-v3
Verification revision
2026-10-28.metric-scope-v3 · 1/1 property · digest 1a346535dfb0
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. Raw HTTP REST/JSON-RPC transport rejection 5/5; raw-number source and conflicting pins are transport-only checks, not engine/SSR input-source parity.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
177/177 golden · 2/2 CVP boundary · 13/13 invalid · 1/1 property · 3/3 metamorphic · 12/12 cross-interface · 1/1 cross-calculator · 6/6 CVP contract · Manifest
Sources
Sources
Evidence
5 legacy golden · 2 legacy boundary · legacy regression suite · 177/177 oracle-backed golden · 13/13 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.1.5 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.

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 1.1.5 · 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: Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.
  • 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
  • NIST DLMF §3.5 — Quadrature — §3.5
    Supports: Trapezoidal, Simpson, and Gaussian quadrature formulas and their assumptions.
Calculation version
1.1.5

Background

Interpretation and common distinctions.

What does this definite integral calculator compute?

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.

The published formula or method is composite trapezoid / Simpson 1/3 / Gauss–Legendre. 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 v1.1.5 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.

Input precision, complete results, and privacy

Inputs are finite decimal numbers or decimal text; a nonzero decimal below binary64 range is rejected, never silently changed to zero. Polynomial input is 2–9 constant-first coefficients; a constant uses c0,0. Method/function aliases must agree. n is an integer: 2–10000 panels for Simpson/trapezoid or 2–8 Gauss nodes. An odd Simpson count advances to the next even count, which is shown in the result. Missing n uses 100 panels or 5 Gauss nodes; explicitly empty inputs remain errors.

The output includes the chosen method, function, full polynomial coefficients, bounds, effective count, approximation, binary64 analytic baseline, signed raw difference, and formula. The historical field abs_err means approximation minus baseline, before 12 significant digit publication; it is not an absolute error or certified bound. This is a fixed rule, not adaptive integration. Sample range failures, unresolved nonzero underflow, and severe cancellation are rejected; split the interval or rescale the function. Figures may be omitted at extreme scales while numeric results remain available.

Normal form calculations run locally without an account. Copy JSON and CSV retain complete inputs, results, and calculation version. A shared query URL sends its parameters to the server when opened and may appear in server or third-party logs. Function and Numerical Workspace links use a fragment and recompute the inputs; stored result claims are ignored. Version notices identify current computation, and machine version pins reject unavailable versions.

Sine bounds are limited to magnitude 1e6 (NUMERICAL_RANGE_ERROR beyond it) because a binary64 fixed grid loses phase resolution for very large arguments. Binary64 node rounding and cancellation near zeros remain precision limitations; the 12 digit publication format is not a uniform accuracy guarantee. A true zero from the selected quadrature rule is allowed even when the analytic baseline is nonzero.

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 Solve a polynomial or sin x for one approximate real zero using Newton, bisection, or secant. Inspect the residual and stopping reason. Calculate 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 one approximate real zero of a polynomial or sin x using Newton, bisection, or secant. Inspect the residual and stopping reason. Calculate 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.