Signed polynomial definite integrals on finite intervals

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

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 — Limits and finite sums are distinct.
• Univariate polynomials over the rationals — Use the restricted polynomial and coefficientwise moments.
• Polynomial antiderivatives and the choice of constant — Check primitive endpoint expressions only within this polynomial moment slice.
• Exact area of half-open rectangles and unit-cell unions — 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.