Question 3801: Three Oriented Polynomial-Moment Integrals

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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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

Step-by-step derivation

S001 · implies

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.

Step identifiers and dependencies

Rule: polynomial_moment_scope · Uses: G001

S002 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

[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 dependencies

Rule: polynomial_primitive_identity · Uses: G001, S002

S004 · implies

Before

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 dependencies

Rule: polynomial_evaluation_order · Uses: G001, S003

S005 · implies

Before

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 dependencies

Rule: polynomial_evaluation_order · Uses: G001, S004

S006 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

[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 dependencies

Rule: polynomial_primitive_identity · Uses: G001, S007

S009 · implies

Before

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 dependencies

Rule: polynomial_evaluation_order · Uses: G001, S008

S010 · implies

Before

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 dependencies

Rule: polynomial_evaluation_order · Uses: G001, S009

S011 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · implies

Before

[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 dependencies

Rule: polynomial_primitive_identity · Uses: G001, S012

S014 · implies

Before

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 dependencies

Rule: polynomial_evaluation_order · Uses: G001, S013

S015 · implies

Before

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 dependencies

Rule: polynomial_evaluation_order · Uses: G001, S014

S016 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

[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 dependencies

Rule: 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

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