# Question 32668: Signed Moment Integral of a Monomial [Return to catalogue page 164](https://ocme.evemisslab.com/en/questions/page/164/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261008-r07-definite-integral-foundations/#q-20261007-advance-20261008-r07-definite-integral-0043) 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. R0064-32668 · q-20261007-advance-20261008-r07-definite-integral-0043 ## Original question and givens On all R, f(x)=(-18)x^5, 0≤n≤5. Use the exact signed moment c(b^(n+1)−a^(n+1))/(n+1) on oriented endpoints a=-3, b=0. Give a reduced fraction with positive denominator, retaining /1 for integers. - G001 — On all R, f(x)=(-18)x^5, 0≤n≤5. Use the exact signed moment c(b^(n+1)−a^(n+1))/(n+1) on oriented endpoints a=-3, b=0. 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-monomial, monomial_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 Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication1 := mul(1, -3) After Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication1 = -3 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 Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication2 := mul(-3, -3) After Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication2 = 9 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 · calculationBefore Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication3 := mul(9, -3) After Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication3 = -27 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, S003 S005 · calculationBefore Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication4 := mul(-27, -3) After Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication4 = 81 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, S004 S006 · calculationBefore Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication5 := mul(81, -3) After Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication5 = -243 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, S005 S007 · calculationBefore Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication6 := mul(-243, -3) After Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication6 = 729 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, S006 S008 · impliesBefore source roles = [Monomial ORIGINAL term lower endpoint, -3, 6] After derived = [729] Conditions: none Show EVERY integer-power multiplication,retaining parity signs for negative bases.Derived values are not RAW-SAFE recapped or computed by floating pow. Step identifiers and dependenciesRule: Moment_INTEGER_power_complete_factors_NO_raw_recap · Uses: G001, S007 S009 · calculationBefore Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication1 := mul(1, 0) After Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication1 = 0 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, S008 S010 · calculationBefore Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication2 := mul(0, 0) After Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication2 = 0 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, S009 S011 · calculationBefore Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication3 := mul(0, 0) After Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication3 = 0 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, S010 S012 · calculationBefore Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication4 := mul(0, 0) After Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication4 = 0 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, S011 S013 · calculationBefore Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication5 := mul(0, 0) After Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication5 = 0 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, S012 S014 · calculationBefore Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication6 := mul(0, 0) After Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication6 = 0 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, S013 S015 · impliesBefore source roles = [Monomial ORIGINAL term upper endpoint, 0, 6] After derived = [0] Conditions: none Show EVERY integer-power multiplication,retaining parity signs for negative bases.Derived values are not RAW-SAFE recapped or computed by floating pow. Step identifiers and dependenciesRule: Moment_INTEGER_power_complete_factors_NO_raw_recap · Uses: G001, S014 S016 · calculationBefore Monomial ORIGINAL term ORIGINAL upper power minus lower power := sub(0, 729) After Monomial ORIGINAL term ORIGINAL upper power minus lower power = -729 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, S015 S017 · calculationBefore Monomial ORIGINAL term ORIGINAL coefficient times power difference := mul(-18, -729) After Monomial ORIGINAL term ORIGINAL coefficient times power difference = 13122 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, S016 S018 · calculationBefore Monomial ORIGINAL term ORIGINAL signed moment over positive exponent := div(13122, 6) After Monomial ORIGINAL term ORIGINAL signed moment over positive exponent = 2187 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, S017 S019 · impliesBefore source roles = [Monomial ORIGINAL term, -18, 5, -3, 0] After derived = [729, 0, -729, 2187/1] Conditions: none Use the DECLARED signed polynomial moment c_k(b^(k+1)−a^(k+1))/(k+1)exactly.Reversed endpoints are legal;do not take absolute values.This restricted moment model is not general FTC/integrability Evidence. Step identifiers and dependenciesRule: Moment_GIVEN_signed_polynomial_coefficient_moment_orientation · Uses: G001, S018 ## Reverse checks - S019 · 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. - S018 · premise_and_result_audit — Recompute Monomial ORIGINAL term ORIGINAL signed moment over positive exponent with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Monomial ORIGINAL term ORIGINAL coefficient times power difference with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Monomial ORIGINAL term ORIGINAL upper power minus lower power with exact rational arithmetic, not rounded decimals. - S015 · 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. - S014 · premise_and_result_audit — Recompute Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication6 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication5 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication4 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication3 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals. - S008 · 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. - S007 · premise_and_result_audit — Recompute Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication6 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication5 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication4 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication3 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication1 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 2187/1 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: 3f3967bf87e2c3bc2cf81184af6152e6c3ae3c78fa845cfdf14a25bd3cea53a1 - candidate record SHA-256: a47b7b04b203b6679d3519a140f6ae61f6bce8473c031ca938f201d84d618e4b - 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-0043.json)