# Question 32638: Signed Moment Integral of a Constant

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0064-32638 · q-20261007-advance-20261008-r07-definite-integral-0013



## Original question and givens

On all R, f(x)=12. Use the scoped signed polynomial moments to find I(f,0,-13), with oriented endpoints. Give a reduced fraction with positive denominator, retaining /1 for integers.


- G001 — On all R, f(x)=12. Use the scoped signed polynomial moments to find I(f,0,-13), with oriented endpoints. Give a reduced fraction with positive denominator, 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](https://ocme.evemisslab.com/en/mko/mko-definite-integral/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-integral-constant, constant_integral, reals, []]

After

derived = [[signed_moment_fraction_RETAIN1]]

Conditions: none

Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status.

Step identifiers and dependenciesRule: R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles · Uses: G001





S002 · calculationBefore

Constant moment ORIGINAL oriented width := sub(-13, 0)

After

Constant moment ORIGINAL oriented width = -13

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

Constant moment ORIGINAL c times oriented width := mul(12, -13)

After

Constant moment ORIGINAL c times oriented width = -156

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

source roles = [12, 0, -13]

After

derived = [-13, -156/1]

Conditions: none

The signed constant moment c(b−a)retains orientation,reverses sign with endpoints,and vanishes at equal endpoints.It is not automatically nonnegative area;integer answers retain/1.

Step identifiers and dependenciesRule: Moment_CONSTANT_oriented_c_width_RETAIN1_not_geometric_area · Uses: G001, S003









## Reverse checks


- S004 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
- S003 · premise_and_result_audit — Recompute Constant moment ORIGINAL c times oriented width with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute Constant moment ORIGINAL oriented width with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.





## Result

-156/1

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: 5a52fa992d599dd1e0213f90f5156a62b245c63d7e712be982b63d79ff28b5d8
- candidate record SHA-256: e823588ff741e855a21bb063eff8222a830e8b59cb52ebaa1af053cff9a79759
- semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b
- question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261008-r07-definite-integral-0013.json)