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Definite Integral Calculator

Composite trapezoid or Simpson 1/3 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 Engine tested · Specification checked · 9/9 tests · Production surface contract 4/4 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
Definite integral of poly, sin, or exp on [a, b].
Scope
Not indefinite CAS
Verification
Engine tested · 9/9 tests · Production surface contract 4/4 · Specification checked · v1.0.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
3 golden · 1 boundary · 5 property · Production surface contract 4/4 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status PASS (0 stale; 164 CURRENT) @ 2026-09-16T06:44:24.909Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

Trapezoidh/2 · (y₀ + 2Σ yᵢ + yₙ)
Simpson 1/3h/3 · (y₀ + 4Σ odd + 2Σ even + yₙ)
iNot indefinite integration and not symbolic simplify. Polynomial coefficients are constant-first. n is the panel count (2–10000).

How to use

1

Choose Simpson or trapezoid

Poly needs coeffs. sin and exp are f(x)=sin x and e^x.

2

Enter [a, b] and panels n

Simpson n is forced even. 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 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
The definite integral ∫_a^b f(x) dx is approximated by composite trapezoid or Simpson 1/3. 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].
Inputs
  • method
  • integrand
  • a
  • b
  • n?
  • coeffs?
Outputs
  • value
  • exact
  • abs_err
Formula
composite trapezoid / Simpson 1/3
Assumptions
  • Not indefinite CAS
  • Polynomial degree ≤ 8
Units
  • dimensionless
Boundary conditions
  • n not in 2–10000 → INVALID_INPUT
  • unknown method (including romberg) → INVALID_MODE
Example
simpson poly 0,1 on [0,1] → 0.5
Validation cases

2 published on this page · 9/9 tests · Production surface contract 4/4 · View evidence

  • method=simpson integrand=poly coeffs=0,1 a=0 b=1 n=10 → exact=0.5
  • method=romberg integrand=sin a=0 b=1 → INVALID_MODE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.0.0

Other calculators in this family: Numerical root, Sine, Exponent .

f(x)=0 Numerical root › sin Sine › x^y Exponent ›
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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.