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R0006-03802 · q-20261007-advance-20261009-r06-definite-integral-0002
On all R, f has ascending coefficients [3], constant term first. For arbitrary oriented endpoints a=3,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 [3], constant term first. For arbitrary oriented endpoints a=3,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.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.
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.
Rule: polynomial_moment_scope · Uses: G001
Before
I(a,b) primitive coefficient1 := div(3, 1)
After
I(a,b) primitive coefficient1 = 3
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
[0, 3]
After
H′=the supplied polynomial;integrate from 3 to -1
Conditions: none
Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference.
Rule: polynomial_primitive_identity · Uses: G001, S002
Before
I(a,b) H(b) ascending coefficients=[0, 3]
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.
Rule: polynomial_evaluation_order · Uses: G001, S003
Before
I(a,b) H(b) Horner multiply before degree0 := mul(3, -1)
After
I(a,b) H(b) Horner multiply before degree0 = -3
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
I(a,b) H(b) Horner add degree0 := sum(-3, 0)
After
I(a,b) H(b) Horner add degree0 = -3
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
I(a,b) H(a) ascending coefficients=[0, 3]
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.
Rule: polynomial_evaluation_order · Uses: G001, S006
Before
I(a,b) H(a) Horner multiply before degree0 := mul(3, 3)
After
I(a,b) H(a) Horner multiply before degree0 = 9
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
I(a,b) H(a) Horner add degree0 := sum(9, 0)
After
I(a,b) H(a) Horner add degree0 = 9
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
I(a,b) oriented moment := sub(-3, 9)
After
I(a,b) oriented moment = -12
Conditions: none
Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
I(b,c) primitive coefficient1 := div(3, 1)
After
I(b,c) primitive coefficient1 = 3
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
[0, 3]
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.
Rule: polynomial_primitive_identity · Uses: G001, S011
Before
I(b,c) H(b) ascending coefficients=[0, 3]
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.
Rule: polynomial_evaluation_order · Uses: G001, S012
Before
I(b,c) H(b) Horner multiply before degree0 := mul(3, 2)
After
I(b,c) H(b) Horner multiply before degree0 = 6
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
I(b,c) H(b) Horner add degree0 := sum(6, 0)
After
I(b,c) H(b) Horner add degree0 = 6
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
I(b,c) H(a) ascending coefficients=[0, 3]
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.
Rule: polynomial_evaluation_order · Uses: G001, S015
Before
I(b,c) H(a) Horner multiply before degree0 := mul(3, -1)
After
I(b,c) H(a) Horner multiply before degree0 = -3
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
I(b,c) H(a) Horner add degree0 := sum(-3, 0)
After
I(b,c) H(a) Horner add degree0 = -3
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
I(b,c) oriented moment := sub(6, -3)
After
I(b,c) oriented moment = 9
Conditions: none
Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
I(a,c) primitive coefficient1 := div(3, 1)
After
I(a,c) primitive coefficient1 = 3
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
[0, 3]
After
H′=the supplied polynomial;integrate from 3 to 2
Conditions: none
Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference.
Rule: polynomial_primitive_identity · Uses: G001, S020
Before
I(a,c) H(b) ascending coefficients=[0, 3]
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.
Rule: polynomial_evaluation_order · Uses: G001, S021
Before
I(a,c) H(b) Horner multiply before degree0 := mul(3, 2)
After
I(a,c) H(b) Horner multiply before degree0 = 6
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
I(a,c) H(b) Horner add degree0 := sum(6, 0)
After
I(a,c) H(b) Horner add degree0 = 6
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
I(a,c) H(a) ascending coefficients=[0, 3]
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.
Rule: polynomial_evaluation_order · Uses: G001, S024
Before
I(a,c) H(a) Horner multiply before degree0 := mul(3, 3)
After
I(a,c) H(a) Horner multiply before degree0 = 9
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
I(a,c) H(a) Horner add degree0 := sum(9, 0)
After
I(a,c) H(a) Horner add degree0 = 9
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
I(a,c) oriented moment := sub(6, 9)
After
I(a,c) oriented moment = -3
Conditions: none
Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
adjacent oriented moment sum := sum(-12, 9)
After
adjacent oriented moment sum = -3
Conditions: none
H(b) cancels between endpoint differences,so a≤b≤c is not required.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
[H(b)−H(a)]+[H(c)−H(b)]
After
H(c)−H(a)=-3
Conditions: none
This algebraic cancellation works for any endpoint order;negative integrals are not replaced by positive area.
Rule: oriented_integral_additivity · Uses: G001, S029
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 adjacent oriented moment sum with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Recompute I(a,c) oriented moment with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Recompute I(a,c) H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Recompute I(a,c) H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.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 I(a,c) H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute I(a,c) H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 I(a,c) primitive coefficient1 with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute I(b,c) oriented moment with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute I(b,c) H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute I(b,c) H(a) Horner multiply before degree0 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(b,c) H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute I(b,c) H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · premise_and_result_audit — Recompute I(b,c) primitive coefficient1 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute I(a,b) oriented moment with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute I(a,b) H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute I(a,b) H(a) Horner multiply before degree0 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 — Recompute I(a,b) H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute I(a,b) H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.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.(-12/1, 9/1, -3/1)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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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