Integral Calculator (symbolic + numeric) mechanism

Model scope

The stated Simpson method and the symbolic antiderivative examples are mathematically correct under the documented assumptions, but the specification needs review because Simpson constraints, error estimation, signed-versus-unsigned output, and the scope of the claimed always-available numeric fallback are not fully defined.

• Simpson's rule is correct only with equally spaced samples, n subintervals, h=(b-a)/n, and an even positive integer n; these constraints are not stated.
• The required numeric error-estimation method is unspecified. A single Simpson evaluation does not itself provide an error estimate; Richardson comparison or another documented method is needed.
• The claim that numeric fallback is always available is too broad unless the implementation specifies handling for singularities, discontinuities, divergent integrals, overflow, and non-finite function values.
• The output says numeric area, but a definite integral is generally a signed quantity; the instrument must distinguish signed integral from geometric area.
• The symbolic method is described as nerdamer/math.js rules rather than a defined algorithm, so branch conventions, constants of integration, special functions, and unsupported forms are ambiguous.

Units and conventions

• Bounds must use the same unit, and the integrand's output unit is function-unit multiplied by bound-unit.
• Polynomial and exponential test functions treat x as dimensionless unless a scale constant is explicitly supplied.
• Trigonometric functions use radians; the zero-boundary vector therefore uses radians.
• Definite integrals below are signed integrals, not absolute geometric areas.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "function": "3*x^2 - 4*x + 1",
    "variable": "x",
    "bounds": null
  },
  "expected_outputs": {
    "antiderivative": "x^3 - 2*x^2 + x + C",
    "spot_checks": [
      {
        "at": 1,
        "input_side": 0,
        "answer_side": 0
      },
      {
        "at": 2,
        "input_side": 5,
        "answer_side": 5
      }
    ]
  },
  "reference_data": {
    "method": "Differentiate the proposed antiderivative and compare with the integrand."
  },
  "working": "Let F(x)=x^3-2*x^2+x. Then F'(x)=3*x^2-4*x+1. At x=1: original=3(1)^2-4(1)+1=0; answer-side derivative=3(1)^2-4(1)+1=0. At x=2: original=3(2)^2-4(2)+1=12-8+1=5; answer-side derivative=12-8+1=5."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The input 1/x over [-1,1] forces the quadrature to evaluate 1/0 and the ordinary definite integral is undefined. A calculator that returns a finite number, such as the Cauchy principal value 0, without explicitly identifying the principal-value convention would be incorrect.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche8/tranche8-results.json
• BOUNDARY_HAZARD_CATALOG.json