# Question 3801: Three Oriented Polynomial-Moment Integrals [Return to catalogue page 20](https://ocme.evemisslab.com/en/questions/page/20/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261009-r06-definite-integral-foundations/#q-20261007-advance-20261009-r06-definite-integral-0001) 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. R0006-03801 · q-20261007-advance-20261009-r06-definite-integral-0001 ## Original question and givens On all R, f has ascending coefficients [0], constant term first. For arbitrary oriented endpoints a=0,b=1,c=2, give (I(f,a,b),I(f,b,c),I(f,a,c)) using the scoped polynomial moments. Use reduced positive-denominator fractions, retaining /1 for integers. - G001 — On all R, f has ascending coefficients [0], constant term first. For arbitrary oriented endpoints a=0,b=1,c=2, give (I(f,a,b),I(f,b,c),I(f,a,c)) using the scoped polynomial moments. 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](https://ocme.evemisslab.com/en/mko/mko-definite-integral/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: polynomial_moment_scope · Uses: G001 S002 · calculationBefore I(a,b) primitive coefficient1 := div(0, 1) After I(a,b) primitive coefficient1 = 0 Conditions: none The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore [0, 0] After H′=the supplied polynomial;integrate from 0 to 1 Conditions: none Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference. Step identifiers and dependenciesRule: polynomial_primitive_identity · Uses: G001, S002 S004 · impliesBefore I(a,b) H(b) ascending coefficients=[0, 0] 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 dependenciesRule: polynomial_evaluation_order · Uses: G001, S003 S005 · impliesBefore I(a,b) H(a) ascending coefficients=[0, 0] 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 dependenciesRule: polynomial_evaluation_order · Uses: G001, S004 S006 · calculationBefore I(a,b) oriented moment := sub(0, 0) After I(a,b) oriented moment = 0 Conditions: none Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore I(b,c) primitive coefficient1 := div(0, 1) After I(b,c) primitive coefficient1 = 0 Conditions: none The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore [0, 0] After H′=the supplied polynomial;integrate from 1 to 2 Conditions: none Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference. Step identifiers and dependenciesRule: polynomial_primitive_identity · Uses: G001, S007 S009 · impliesBefore I(b,c) H(b) ascending coefficients=[0, 0] After evaluate at x=2 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 dependenciesRule: polynomial_evaluation_order · Uses: G001, S008 S010 · impliesBefore I(b,c) H(a) ascending coefficients=[0, 0] 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 dependenciesRule: polynomial_evaluation_order · Uses: G001, S009 S011 · calculationBefore I(b,c) oriented moment := sub(0, 0) After I(b,c) oriented moment = 0 Conditions: none Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore I(a,c) primitive coefficient1 := div(0, 1) After I(a,c) primitive coefficient1 = 0 Conditions: none The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore [0, 0] After H′=the supplied polynomial;integrate from 0 to 2 Conditions: none Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference. Step identifiers and dependenciesRule: polynomial_primitive_identity · Uses: G001, S012 S014 · impliesBefore I(a,c) H(b) ascending coefficients=[0, 0] After evaluate at x=2 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 dependenciesRule: polynomial_evaluation_order · Uses: G001, S013 S015 · impliesBefore I(a,c) H(a) ascending coefficients=[0, 0] 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 dependenciesRule: polynomial_evaluation_order · Uses: G001, S014 S016 · calculationBefore I(a,c) oriented moment := sub(0, 0) After I(a,c) oriented moment = 0 Conditions: none Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore adjacent oriented moment sum := sum(0, 0) After adjacent oriented moment sum = 0 Conditions: none H(b) cancels between endpoint differences,so a≤b≤c is not required. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore [H(b)−H(a)]+[H(c)−H(b)] After H(c)−H(a)=0 Conditions: none This algebraic cancellation works for any endpoint order;negative integrals are not replaced by positive area. Step identifiers and dependenciesRule: oriented_integral_additivity · Uses: G001, S017 ## Reverse checks - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Recompute adjacent oriented moment sum with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute I(a,c) oriented moment 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 — 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 I(a,c) primitive coefficient1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute I(b,c) oriented moment with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute I(b,c) primitive coefficient1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute I(a,b) oriented moment with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Recompute I(a,b) primitive coefficient1 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 (0/1, 0/1, 0/1) 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: f86bb5a8148a1fdaef7e77dfda9141f197edca1783dc59122b363d847552d961 - candidate record SHA-256: 93b64a9631b5e5a981171c27fe40bd30694d604a567d719a9997f6e8f2878888 - semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d - question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261009-r06-definite-integral-0001.json)