# Question 3817: 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-0017) 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. R0006-03817 · q-20261007-advance-20261009-r06-definite-integral-0017 ## Original question and givens On all R, f has ascending coefficients [9, 17, 18, 1], constant term first. For arbitrary oriented endpoints a=-1,b=0,c=3, 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 [9, 17, 18, 1], constant term first. For arbitrary oriented endpoints a=-1,b=0,c=3, 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(9, 1) After I(a,b) primitive coefficient1 = 9 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 · calculationBefore I(a,b) primitive coefficient2 := div(17, 2) After I(a,b) primitive coefficient2 = 17/2 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, S002 S004 · calculationBefore I(a,b) primitive coefficient3 := div(18, 3) After I(a,b) primitive coefficient3 = 6 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, S003 S005 · calculationBefore I(a,b) primitive coefficient4 := div(1, 4) After I(a,b) primitive coefficient4 = 1/4 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, S004 S006 · impliesBefore [0, 9, 17/2, 6, 1/4] After H′=the supplied polynomial;integrate from -1 to 0 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, S005 S007 · impliesBefore I(a,b) H(b) ascending coefficients=[0, 9, 17/2, 6, 1/4] 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, S006 S008 · calculationBefore I(a,b) H(b) Horner multiply before degree3 := mul(1/4, 0) After I(a,b) H(b) Horner multiply before degree3 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore I(a,b) H(b) Horner add degree3 := sum(0, 6) After I(a,b) H(b) Horner add degree3 = 6 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore I(a,b) H(b) Horner multiply before degree2 := mul(6, 0) After I(a,b) H(b) Horner multiply before degree2 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore I(a,b) H(b) Horner add degree2 := sum(0, 17/2) After I(a,b) H(b) Horner add degree2 = 17/2 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore I(a,b) H(b) Horner multiply before degree1 := mul(17/2, 0) After I(a,b) H(b) Horner multiply before degree1 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore I(a,b) H(b) Horner add degree1 := sum(0, 9) After I(a,b) H(b) Horner add degree1 = 9 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore I(a,b) H(b) Horner multiply before degree0 := mul(9, 0) After I(a,b) H(b) Horner multiply before degree0 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore I(a,b) H(b) Horner add degree0 := sum(0, 0) After I(a,b) H(b) Horner add degree0 = 0 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore I(a,b) H(a) ascending coefficients=[0, 9, 17/2, 6, 1/4] 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, S015 S017 · calculationBefore I(a,b) H(a) Horner multiply before degree3 := mul(1/4, -1) After I(a,b) H(a) Horner multiply before degree3 = -1/4 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore I(a,b) H(a) Horner add degree3 := sum(-1/4, 6) After I(a,b) H(a) Horner add degree3 = 23/4 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore I(a,b) H(a) Horner multiply before degree2 := mul(23/4, -1) After I(a,b) H(a) Horner multiply before degree2 = -23/4 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore I(a,b) H(a) Horner add degree2 := sum(-23/4, 17/2) After I(a,b) H(a) Horner add degree2 = 11/4 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore I(a,b) H(a) Horner multiply before degree1 := mul(11/4, -1) After I(a,b) H(a) Horner multiply before degree1 = -11/4 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore I(a,b) H(a) Horner add degree1 := sum(-11/4, 9) After I(a,b) H(a) Horner add degree1 = 25/4 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore I(a,b) H(a) Horner multiply before degree0 := mul(25/4, -1) After I(a,b) H(a) Horner multiply before degree0 = -25/4 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore I(a,b) H(a) Horner add degree0 := sum(-25/4, 0) After I(a,b) H(a) Horner add degree0 = -25/4 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore I(a,b) oriented moment := sub(0, -25/4) After I(a,b) oriented moment = 25/4 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, S024 S026 · calculationBefore I(b,c) primitive coefficient1 := div(9, 1) After I(b,c) primitive coefficient1 = 9 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, S025 S027 · calculationBefore I(b,c) primitive coefficient2 := div(17, 2) After I(b,c) primitive coefficient2 = 17/2 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, S026 S028 · calculationBefore I(b,c) primitive coefficient3 := div(18, 3) After I(b,c) primitive coefficient3 = 6 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, S027 S029 · calculationBefore I(b,c) primitive coefficient4 := div(1, 4) After I(b,c) primitive coefficient4 = 1/4 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, S028 S030 · impliesBefore [0, 9, 17/2, 6, 1/4] 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 dependenciesRule: polynomial_primitive_identity · Uses: G001, S029 S031 · impliesBefore I(b,c) H(b) ascending coefficients=[0, 9, 17/2, 6, 1/4] 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 dependenciesRule: polynomial_evaluation_order · Uses: G001, S030 S032 · calculationBefore I(b,c) H(b) Horner multiply before degree3 := mul(1/4, 3) After I(b,c) H(b) Horner multiply before degree3 = 3/4 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore I(b,c) H(b) Horner add degree3 := sum(3/4, 6) After I(b,c) H(b) Horner add degree3 = 27/4 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore I(b,c) H(b) Horner multiply before degree2 := mul(27/4, 3) After I(b,c) H(b) Horner multiply before degree2 = 81/4 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore I(b,c) H(b) Horner add degree2 := sum(81/4, 17/2) After I(b,c) H(b) Horner add degree2 = 115/4 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore I(b,c) H(b) Horner multiply before degree1 := mul(115/4, 3) After I(b,c) H(b) Horner multiply before degree1 = 345/4 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore I(b,c) H(b) Horner add degree1 := sum(345/4, 9) After I(b,c) H(b) Horner add degree1 = 381/4 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore I(b,c) H(b) Horner multiply before degree0 := mul(381/4, 3) After I(b,c) H(b) Horner multiply before degree0 = 1143/4 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore I(b,c) H(b) Horner add degree0 := sum(1143/4, 0) After I(b,c) H(b) Horner add degree0 = 1143/4 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · impliesBefore I(b,c) H(a) ascending coefficients=[0, 9, 17/2, 6, 1/4] 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, S039 S041 · calculationBefore I(b,c) H(a) Horner multiply before degree3 := mul(1/4, 0) After I(b,c) H(a) Horner multiply before degree3 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore I(b,c) H(a) Horner add degree3 := sum(0, 6) After I(b,c) H(a) Horner add degree3 = 6 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore I(b,c) H(a) Horner multiply before degree2 := mul(6, 0) After I(b,c) H(a) Horner multiply before degree2 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore I(b,c) H(a) Horner add degree2 := sum(0, 17/2) After I(b,c) H(a) Horner add degree2 = 17/2 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore I(b,c) H(a) Horner multiply before degree1 := mul(17/2, 0) After I(b,c) H(a) Horner multiply before degree1 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore I(b,c) H(a) Horner add degree1 := sum(0, 9) After I(b,c) H(a) Horner add degree1 = 9 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore I(b,c) H(a) Horner multiply before degree0 := mul(9, 0) After I(b,c) H(a) Horner multiply before degree0 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · calculationBefore I(b,c) H(a) Horner add degree0 := sum(0, 0) After I(b,c) H(a) Horner add degree0 = 0 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · calculationBefore I(b,c) oriented moment := sub(1143/4, 0) After I(b,c) oriented moment = 1143/4 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, S048 S050 · calculationBefore I(a,c) primitive coefficient1 := div(9, 1) After I(a,c) primitive coefficient1 = 9 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, S049 S051 · calculationBefore I(a,c) primitive coefficient2 := div(17, 2) After I(a,c) primitive coefficient2 = 17/2 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, S050 S052 · calculationBefore I(a,c) primitive coefficient3 := div(18, 3) After I(a,c) primitive coefficient3 = 6 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, S051 S053 · calculationBefore I(a,c) primitive coefficient4 := div(1, 4) After I(a,c) primitive coefficient4 = 1/4 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, S052 S054 · impliesBefore [0, 9, 17/2, 6, 1/4] After H′=the supplied polynomial;integrate from -1 to 3 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, S053 S055 · impliesBefore I(a,c) H(b) ascending coefficients=[0, 9, 17/2, 6, 1/4] 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 dependenciesRule: polynomial_evaluation_order · Uses: G001, S054 S056 · calculationBefore I(a,c) H(b) Horner multiply before degree3 := mul(1/4, 3) After I(a,c) H(b) Horner multiply before degree3 = 3/4 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S055 S057 · calculationBefore I(a,c) H(b) Horner add degree3 := sum(3/4, 6) After I(a,c) H(b) Horner add degree3 = 27/4 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S056 S058 · calculationBefore I(a,c) H(b) Horner multiply before degree2 := mul(27/4, 3) After I(a,c) H(b) Horner multiply before degree2 = 81/4 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S057 S059 · calculationBefore I(a,c) H(b) Horner add degree2 := sum(81/4, 17/2) After I(a,c) H(b) Horner add degree2 = 115/4 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S058 S060 · calculationBefore I(a,c) H(b) Horner multiply before degree1 := mul(115/4, 3) After I(a,c) H(b) Horner multiply before degree1 = 345/4 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S059 S061 · calculationBefore I(a,c) H(b) Horner add degree1 := sum(345/4, 9) After I(a,c) H(b) Horner add degree1 = 381/4 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S060 S062 · calculationBefore I(a,c) H(b) Horner multiply before degree0 := mul(381/4, 3) After I(a,c) H(b) Horner multiply before degree0 = 1143/4 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S061 S063 · calculationBefore I(a,c) H(b) Horner add degree0 := sum(1143/4, 0) After I(a,c) H(b) Horner add degree0 = 1143/4 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S062 S064 · impliesBefore I(a,c) H(a) ascending coefficients=[0, 9, 17/2, 6, 1/4] 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, S063 S065 · calculationBefore I(a,c) H(a) Horner multiply before degree3 := mul(1/4, -1) After I(a,c) H(a) Horner multiply before degree3 = -1/4 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S064 S066 · calculationBefore I(a,c) H(a) Horner add degree3 := sum(-1/4, 6) After I(a,c) H(a) Horner add degree3 = 23/4 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S065 S067 · calculationBefore I(a,c) H(a) Horner multiply before degree2 := mul(23/4, -1) After I(a,c) H(a) Horner multiply before degree2 = -23/4 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S066 S068 · calculationBefore I(a,c) H(a) Horner add degree2 := sum(-23/4, 17/2) After I(a,c) H(a) Horner add degree2 = 11/4 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067 S069 · calculationBefore I(a,c) H(a) Horner multiply before degree1 := mul(11/4, -1) After I(a,c) H(a) Horner multiply before degree1 = -11/4 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S068 S070 · calculationBefore I(a,c) H(a) Horner add degree1 := sum(-11/4, 9) After I(a,c) H(a) Horner add degree1 = 25/4 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S069 S071 · calculationBefore I(a,c) H(a) Horner multiply before degree0 := mul(25/4, -1) After I(a,c) H(a) Horner multiply before degree0 = -25/4 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S070 S072 · calculationBefore I(a,c) H(a) Horner add degree0 := sum(-25/4, 0) After I(a,c) H(a) Horner add degree0 = -25/4 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S071 S073 · calculationBefore I(a,c) oriented moment := sub(1143/4, -25/4) After I(a,c) oriented moment = 292 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, S072 S074 · calculationBefore adjacent oriented moment sum := sum(25/4, 1143/4) After adjacent oriented moment sum = 292 Conditions: none H(b) cancels between endpoint differences,so a≤b≤c is not required. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S073 S075 · impliesBefore [H(b)−H(a)]+[H(c)−H(b)] After H(c)−H(a)=292 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, S074 ## Reverse checks - S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S074 · premise_and_result_audit — Recompute adjacent oriented moment sum with exact rational arithmetic, not rounded decimals. - S073 · premise_and_result_audit — Recompute I(a,c) oriented moment with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Recompute I(a,c) H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Recompute I(a,c) H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S070 · premise_and_result_audit — Recompute I(a,c) H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals. - S069 · premise_and_result_audit — Recompute I(a,c) H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals. - S068 · premise_and_result_audit — Recompute I(a,c) H(a) Horner add degree2 with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Recompute I(a,c) H(a) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals. - S066 · premise_and_result_audit — Recompute I(a,c) H(a) Horner add degree3 with exact rational arithmetic, not rounded decimals. - S065 · premise_and_result_audit — Recompute I(a,c) H(a) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals. - S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S063 · premise_and_result_audit — Recompute I(a,c) H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals. - S062 · premise_and_result_audit — Recompute I(a,c) H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Recompute I(a,c) H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals. - S060 · premise_and_result_audit — Recompute I(a,c) H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Recompute I(a,c) H(b) Horner add degree2 with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Recompute I(a,c) H(b) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute I(a,c) H(b) Horner add degree3 with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute I(a,c) H(b) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S053 · premise_and_result_audit — Recompute I(a,c) primitive coefficient4 with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute I(a,c) primitive coefficient3 with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute I(a,c) primitive coefficient2 with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute I(a,c) primitive coefficient1 with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute I(b,c) oriented moment with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute I(b,c) H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute I(b,c) H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute I(b,c) H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute I(b,c) H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute I(b,c) H(a) Horner add degree2 with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute I(b,c) H(a) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute I(b,c) H(a) Horner add degree3 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute I(b,c) H(a) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S039 · premise_and_result_audit — Recompute I(b,c) H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute I(b,c) H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute I(b,c) H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute I(b,c) H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute I(b,c) H(b) Horner add degree2 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute I(b,c) H(b) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute I(b,c) H(b) Horner add degree3 with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute I(b,c) H(b) Horner multiply before degree3 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S029 · premise_and_result_audit — Recompute I(b,c) primitive coefficient4 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute I(b,c) primitive coefficient3 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute I(b,c) primitive coefficient2 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute I(b,c) primitive coefficient1 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute I(a,b) oriented moment with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute I(a,b) H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute I(a,b) H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute I(a,b) H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute I(a,b) H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute I(a,b) H(a) Horner add degree2 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute I(a,b) H(a) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute I(a,b) H(a) Horner add degree3 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute I(a,b) H(a) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Recompute I(a,b) H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute I(a,b) H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute I(a,b) H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute I(a,b) H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute I(a,b) H(b) Horner add degree2 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute I(a,b) H(b) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute I(a,b) H(b) Horner add degree3 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute I(a,b) H(b) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute I(a,b) primitive coefficient4 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute I(a,b) primitive coefficient3 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute I(a,b) primitive coefficient2 with exact rational arithmetic, not rounded decimals. - 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 (25/4, 1143/4, 292/1) 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: 1a599e062281341b9c80bb28a7bca6003336832e47ea6596166019767a9ee63c - candidate record SHA-256: 71f0cfcd3ecfb316d910dd7c460bf544fabe4c1334f5c329ad9cec25089b52bd - 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-0017.json)