Working definition
Numerical integration approximates a definite integral from sampled function values. Accuracy depends on smoothness, interval partition, singularities, oscillation, and the chosen quadrature rule; adaptive methods allocate evaluations according to estimated local error.
Notation
I = ∫ₐᵇ f(x) dxI ≈ Σ wᵢf(xᵢ)Assumptions
- The integral exists under the chosen definition.
- The integrand can be evaluated accurately.
- Discontinuities and singularities are isolated.
Invariants
- Additivity holds across exact subinterval partitions.
- Units equal integrand units times integration-variable units.
- Convergence should stabilize under refinement.
Reproducible procedure
- Partition at known discontinuities.
- Apply an appropriate fixed or adaptive rule.
- Repeat with tighter tolerance or an independent rule.
Error and boundary controls
- Quadrature estimates can fail on unresolved spikes.
- Cancellation can hide large local errors.
- Model and measurement uncertainty remain separate.
What this does not establish
An accurately accumulated exposure or signal does not by itself identify causation or predictive usefulness.
Explicit applications
1 cross-domain bridges
Accumulated thermal exposure
Integrate a time-varying temperature or power profile to compare qualified process exposure.
Inputs
- temperature time series
- time intervals
- response model
Outputs
- integrated exposure
- numerical error estimate
- coverage gaps
Transformation: Apply declared quadrature over valid segments and propagate sensor uncertainty.
Limit: Equal integrated exposure need not imply equal material response when kinetics are nonlinear.
Open connected system →Authoritative references
- [1]DLMF Chapter 3: Numerical Methods · National Institute of Standards and Technology
Establishes: Reference definitions, algorithms, convergence conditions, and error terms for interpolation, quadrature, differentiation, and nonlinear equation solving.
Boundary: A numerical method is reliable only under its stated regularity, conditioning, precision, and convergence assumptions; the reference does not validate any domain interpretation.
- [2]NIST Technical Note 1297: Guidelines for Evaluating and Expressing Measurement Uncertainty · National Institute of Standards and Technology
Establishes: A measurement framework for identifying uncertainty components, combining standard uncertainties, and reporting expanded uncertainty with declared coverage.
Boundary: Reported uncertainty describes the measurement model and included components. It is not a guarantee that all systematic errors or model inadequacies were found.