Question 3852: Exact Moment Integral Minus a Finite Left Sum

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0006-03852 · q-20261007-advance-20261009-r06-definite-integral-0052

Original question and givens

On all R, f(x)=(-2)x+(-11). On a=-1≤b=1, use N=1 uniform left cells. Find the signed error I−S_N, exact moment integral minus the finite left sum. Use reduced positive-denominator fractions, retaining /1 for integers.

G001 — On all R, f(x)=(-2)x+(-11). On a=-1≤b=1, use N=1 uniform left cells. Find the signed error I−S_N, exact moment integral minus the finite left sum. Use reduced positive-denominator fractions, retaining /1 for integers.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

mko-definite-integral

Step-by-step derivation

S001 · implies

Before

explicit real polynomial moment or constant integrand model

After

oriented I(a,b)=H(b)−H(a);do not sort endpoints

Conditions: none

Real polynomials are continuous and integrable;use termwise primitive endpoint differences for exact oriented integrals,not absolute area or a finite sum.

Step identifiers and dependencies

Rule: polynomial_moment_scope · Uses: G001

S002 · calculation

Before

interval width := sub(1, -1)

After

interval width = 2

Conditions: none

Compute the nonnegative interval width,allowing zero width.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

partition step := div(2, 1)

After

partition step = 2

Conditions: none

Divide by positive integer N to get the exact step.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

left sample0 offset := mul(0, 2)

After

left sample0 offset = 0

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

left sample0 x := sum(-1, 0)

After

left sample0 x = -1

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

left sample0 mx := mul(-2, -1)

After

left sample0 mx = 2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

left sample0 value := sum(2, -11)

After

left sample0 value = -9

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

left sample0 rectangle := mul(-9, 2)

After

left sample0 rectangle = -18

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · implies

Before

[-1, -9, -18]

After

finite left-sum term index=0

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependencies

Rule: left_partition_term · Uses: G001, S008

S010 · calculation

Before

finite left sum := sum(-18)

After

finite left sum = -18

Conditions: none

Add all N contributions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

affine integral primitive coefficient1 := div(-11, 1)

After

affine integral primitive coefficient1 = -11

Conditions: none

The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

affine integral primitive coefficient2 := div(-2, 2)

After

affine integral primitive coefficient2 = -1

Conditions: none

The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · implies

Before

[0, -11, -1]

After

H′=the supplied polynomial;integrate from -1 to 1

Conditions: none

Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference.

Step identifiers and dependencies

Rule: polynomial_primitive_identity · Uses: G001, S012

S014 · implies

Before

affine integral H(b) ascending coefficients=[0, -11, -1]

After

evaluate at x=1

Conditions: none

Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list.

Step identifiers and dependencies

Rule: polynomial_evaluation_order · Uses: G001, S013

S015 · calculation

Before

affine integral H(b) Horner multiply before degree1 := mul(-1, 1)

After

affine integral H(b) Horner multiply before degree1 = -1

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

affine integral H(b) Horner add degree1 := sum(-1, -11)

After

affine integral H(b) Horner add degree1 = -12

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

affine integral H(b) Horner multiply before degree0 := mul(-12, 1)

After

affine integral H(b) Horner multiply before degree0 = -12

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

affine integral H(b) Horner add degree0 := sum(-12, 0)

After

affine integral H(b) Horner add degree0 = -12

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · implies

Before

affine integral H(a) ascending coefficients=[0, -11, -1]

After

evaluate at x=-1

Conditions: none

Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list.

Step identifiers and dependencies

Rule: polynomial_evaluation_order · Uses: G001, S018

S020 · calculation

Before

affine integral H(a) Horner multiply before degree1 := mul(-1, -1)

After

affine integral H(a) Horner multiply before degree1 = 1

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

affine integral H(a) Horner add degree1 := sum(1, -11)

After

affine integral H(a) Horner add degree1 = -10

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

affine integral H(a) Horner multiply before degree0 := mul(-10, -1)

After

affine integral H(a) Horner multiply before degree0 = 10

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

affine integral H(a) Horner add degree0 := sum(10, 0)

After

affine integral H(a) Horner add degree0 = 10

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

affine integral oriented moment := sub(-12, 10)

After

affine integral oriented moment = -22

Conditions: none

Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

integral minus left sum := sub(-22, -18)

After

integral minus left sum = -4

Conditions: none

The requested direction is I−S_N;a negative slope may give a negative difference,not absolute error.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · implies

Before

I=-22;S_N=-18;N=1

After

signed finite-sum difference=-4

Conditions: none

This is the exact difference for finite N,not evidence of a general convergence rate or integrability of arbitrary functions.

Step identifiers and dependencies

Rule: finite_sum_vs_integral · Uses: G001, S025

Reverse checks

S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S025 · premise_and_result_audit — Recompute integral minus left sum with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Recompute affine integral oriented moment with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S018 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S012 · premise_and_result_audit — Recompute affine integral primitive coefficient2 with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute affine integral primitive coefficient1 with exact rational arithmetic, not rounded decimals.
S010 · premise_and_result_audit — Recompute finite left sum with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S008 · premise_and_result_audit — Recompute left sample0 rectangle with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute left sample0 value with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Recompute left sample0 mx with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute left sample0 x with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute left sample0 offset with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute partition step with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute interval width with exact rational arithmetic, not rounded decimals.
S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

-4/1

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts

input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: 8951bd89bae50bcab63895be0e40e51f4e1cebf2deca8ec3206fb67aa39faf65
candidate record SHA-256: f106b89800bb53d056d303f0a5d5e3cb9f24208ab34e009866be698cb76a0a72
semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d
question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855
Lean: NOT_RUN · human review: NOT_PERFORMED

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