Question 3853: 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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R0006-03853 · q-20261007-advance-20261009-r06-definite-integral-0053

Original question and givens

On all R, f(x)=(3)x+(-10). On a=0≤b=3, use N=3 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)=(3)x+(-10). On a=0≤b=3, use N=3 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(3, 0)

After

interval width = 3

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(3, 3)

After

partition step = 1

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, 1)

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(0, 0)

After

left sample0 x = 0

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(3, 0)

After

left sample0 mx = 0

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(0, -10)

After

left sample0 value = -10

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(-10, 1)

After

left sample0 rectangle = -10

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

[0, -10, -10]

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

left sample1 offset := mul(1, 1)

After

left sample1 offset = 1

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

left sample1 x := sum(0, 1)

After

left sample1 x = 1

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

left sample1 mx := mul(3, 1)

After

left sample1 mx = 3

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

left sample1 value := sum(3, -10)

After

left sample1 value = -7

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

left sample1 rectangle := mul(-7, 1)

After

left sample1 rectangle = -7

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, S013

S015 · implies

Before

[1, -7, -7]

After

finite left-sum term index=1

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, S014

S016 · calculation

Before

left sample2 offset := mul(2, 1)

After

left sample2 offset = 2

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

left sample2 x := sum(0, 2)

After

left sample2 x = 2

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

left sample2 mx := mul(3, 2)

After

left sample2 mx = 6

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

left sample2 value := sum(6, -10)

After

left sample2 value = -4

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

left sample2 rectangle := mul(-4, 1)

After

left sample2 rectangle = -4

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, S019

S021 · implies

Before

[2, -4, -4]

After

finite left-sum term index=2

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, S020

S022 · calculation

Before

finite left sum := sum(-10, -7, -4)

After

finite left sum = -21

Conditions: none

Add all N contributions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

affine integral primitive coefficient1 := div(-10, 1)

After

affine integral primitive coefficient1 = -10

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, S022

S024 · calculation

Before

affine integral primitive coefficient2 := div(3, 2)

After

affine integral primitive coefficient2 = 3/2

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, S023

S025 · implies

Before

[0, -10, 3/2]

After

H′=the supplied polynomial;integrate from 0 to 3

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, S024

S026 · implies

Before

affine integral H(b) ascending coefficients=[0, -10, 3/2]

After

evaluate at x=3

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, S025

S027 · calculation

Before

affine integral H(b) Horner multiply before degree1 := mul(3/2, 3)

After

affine integral H(b) Horner multiply before degree1 = 9/2

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

affine integral H(b) Horner add degree1 := sum(9/2, -10)

After

affine integral H(b) Horner add degree1 = -11/2

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

affine integral H(b) Horner multiply before degree0 := mul(-11/2, 3)

After

affine integral H(b) Horner multiply before degree0 = -33/2

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

affine integral H(b) Horner add degree0 := sum(-33/2, 0)

After

affine integral H(b) Horner add degree0 = -33/2

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · implies

Before

affine integral H(a) ascending coefficients=[0, -10, 3/2]

After

evaluate at x=0

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, S030

S032 · calculation

Before

affine integral H(a) Horner multiply before degree1 := mul(3/2, 0)

After

affine integral H(a) Horner multiply before degree1 = 0

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

affine integral H(a) Horner add degree1 := sum(0, -10)

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, S032

S034 · calculation

Before

affine integral H(a) Horner multiply before degree0 := mul(-10, 0)

After

affine integral H(a) Horner multiply before degree0 = 0

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

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

After

affine integral H(a) Horner add degree0 = 0

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

affine integral oriented moment := sub(-33/2, 0)

After

affine integral oriented moment = -33/2

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, S035

S037 · calculation

Before

integral minus left sum := sub(-33/2, -21)

After

integral minus left sum = 9/2

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, S036

S038 · implies

Before

I=-33/2;S_N=-21;N=3

After

signed finite-sum difference=9/2

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, S037

Reverse checks

S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S037 · premise_and_result_audit — Recompute integral minus left sum with exact rational arithmetic, not rounded decimals.
S036 · premise_and_result_audit — Recompute affine integral oriented moment with exact rational arithmetic, not rounded decimals.
S035 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.
S034 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S033 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals.
S032 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S030 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.
S029 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S028 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S024 · premise_and_result_audit — Recompute affine integral primitive coefficient2 with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Recompute affine integral primitive coefficient1 with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute finite left sum with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S020 · premise_and_result_audit — Recompute left sample2 rectangle with exact rational arithmetic, not rounded decimals.
S019 · premise_and_result_audit — Recompute left sample2 value with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Recompute left sample2 mx with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Recompute left sample2 x with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Recompute left sample2 offset with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S014 · premise_and_result_audit — Recompute left sample1 rectangle with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Recompute left sample1 value with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Recompute left sample1 mx with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute left sample1 x with exact rational arithmetic, not rounded decimals.
S010 · premise_and_result_audit — Recompute left sample1 offset 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

9/2

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.

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input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: c25c9cac76b29536d5930dc59ea75df1b32b2fb8a36ca7e6610b4949c20ea3ca
candidate record SHA-256: 60df723fde8e9b7773101c1a7d7c095419b29753281907185e56ab99294f5405
semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d
question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855
Lean: NOT_RUN · human review: NOT_PERFORMED

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