# Signed polynomial definite integrals on finite intervals

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Continuous polynomials have oriented definite integrals with exact moment formulas; a finite left Riemann sum is not a general integral or a completed general FTC bridge.

mko-definite-integral · definition · v0.10.0



## Mathematical statement

For a continuous polynomial f on a finite real interval a≤b, its Riemann integral is the common limit of all valid tagged-partition sums as mesh tends to zero. Reverse endpoints by I(f,a,b)=−I(f,b,a), and set I(f,a,a)=0. This slice explicitly uses I(x^n,a,b)=(b^(n+1)−a^(n+1))/(n+1), and for f=Σc_k*x^k, sums these moments coefficientwise.





## Representative formula (exact TeX)

I(f,a,b)





## Assumptions


- Exercise endpoints and coefficients are safe integers. General input polynomials have degree at most three, and monomials c*x^n separately allow n from zero through five. They are continuous on R and integrable on finite intervals.
- The integral is oriented and signed, not necessarily nonnegative. Reverse endpoints change the sign; taking an absolute value does not generally give the stated integral.
- Finite left sums require a≤b and N from one through fifty, with Δx=(b−a)/N and left points i=0 through N−1. A zero-length interval gives zero; a finite sum generally differs from the integral.





## Symbol scope


- I — scoped signed polynomial interval integral
- f — specified continuous real polynomial
- a — oriented starting endpoint
- b — oriented ending endpoint





## Intuition, definition, and misconceptions

Convergence of one particular equal-step left-sum sequence does not establish a Riemann integral for an arbitrary function. The definition controls valid tagged partitions; the exercises use explicitly stated moments of continuous polynomials.

A constant c integrates to c(b−a); monomial and polynomial cases use exact moments in this object. Producer primitive endpoint differences provide another restricted algorithm, while acceptance directly accumulates moments. This is not a completed general fundamental-theorem-of-calculus prerequisite or new formal Evidence.

An affine uniform left sum is computed exactly as Σ(mx_i+c)Δx, x_i=a+iΔx. A producer arithmetic-progression formula is checked by termwise rational accumulation. Equality in a constant special case does not turn a finite sum into a proof of general integration.

Common errors include naming a finite left sum an integral, omitting orientation signs, interpreting every integral as nonnegative area, or calling restricted moment checks a general FTC proof.





## Adopted derivation guide

[Definite integral: computing ∫₀³(x²−4)dx as a limit of left sums and by the moment formula](https://ocme.evemisslab.com/en/derivations/derivation-mko-definite-integral-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Function limits on punctured real domains](https://ocme.evemisslab.com/en/mko/mko-function-limit/) — Limits and finite sums are distinct.
- [Univariate polynomials over the rationals](https://ocme.evemisslab.com/en/mko/mko-polynomial/) — Use the restricted polynomial and coefficientwise moments.
- [Polynomial antiderivatives and the choice of constant](https://ocme.evemisslab.com/en/mko/mko-antiderivative/) — Check primitive endpoint expressions only within this polynomial moment slice.
- [Exact area of half-open rectangles and unit-cell unions](https://ocme.evemisslab.com/en/mko/mko-area/) — Distinguish signed integrals from nonnegative geometric area without inheriting a general area proof.





## Verification boundary

Restricted polynomial moments and finite sums can be recomputed; Riemann/intervalIntegral bridges and general FTC have no attached formal Evidence.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Bridge pinned intervalIntegral.integral_pow and oriented endpoints; Riemann/Lebesgue agreement and general FTC remain independent open obligations.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/definite-integral.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-definite-integral/index.md)