OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 3821: Three Oriented Polynomial-Moment Integrals

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

Return to catalogue page 20

← Question derivation index · Original question batch

R0006-03821 · q-20261007-advance-20261009-r06-definite-integral-0021

Original question and givens

On all R, f has ascending coefficients [13, 21, 22, 1], constant term first. For arbitrary oriented endpoints a=0,b=1,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 [13, 21, 22, 1], constant term first. For arbitrary oriented endpoints a=0,b=1,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.

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(13, 1)

After

I(a,b) primitive coefficient1 = 13

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 · calculation

Before

I(a,b) primitive coefficient2 := div(21, 2)

After

I(a,b) primitive coefficient2 = 21/2

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, S002

S004 · calculation

Before

I(a,b) primitive coefficient3 := div(22, 3)

After

I(a,b) primitive coefficient3 = 22/3

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, S003

S005 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

[0, 13, 21/2, 22/3, 1/4]

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, S005

S007 · implies

Before

I(a,b) H(b) ascending coefficients=[0, 13, 21/2, 22/3, 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 dependencies

Rule: polynomial_evaluation_order · Uses: G001, S006

S008 · calculation

Before

I(a,b) H(b) Horner multiply before degree3 := mul(1/4, 1)

After

I(a,b) H(b) Horner multiply before degree3 = 1/4

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

I(a,b) H(b) Horner add degree3 := sum(1/4, 22/3)

After

I(a,b) H(b) Horner add degree3 = 91/12

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

I(a,b) H(b) Horner multiply before degree2 := mul(91/12, 1)

After

I(a,b) H(b) Horner multiply before degree2 = 91/12

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

I(a,b) H(b) Horner add degree2 := sum(91/12, 21/2)

After

I(a,b) H(b) Horner add degree2 = 217/12

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

I(a,b) H(b) Horner multiply before degree1 := mul(217/12, 1)

After

I(a,b) H(b) Horner multiply before degree1 = 217/12

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

I(a,b) H(b) Horner add degree1 := sum(217/12, 13)

After

I(a,b) H(b) Horner add degree1 = 373/12

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

I(a,b) H(b) Horner multiply before degree0 := mul(373/12, 1)

After

I(a,b) H(b) Horner multiply before degree0 = 373/12

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

I(a,b) H(b) Horner add degree0 := sum(373/12, 0)

After

I(a,b) H(b) Horner add degree0 = 373/12

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

I(a,b) H(a) ascending coefficients=[0, 13, 21/2, 22/3, 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 dependencies

Rule: polynomial_evaluation_order · Uses: G001, S015

S017 · calculation

Before

I(a,b) H(a) Horner multiply before degree3 := mul(1/4, 0)

After

I(a,b) H(a) Horner multiply before degree3 = 0

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

I(a,b) H(a) Horner add degree3 := sum(0, 22/3)

After

I(a,b) H(a) Horner add degree3 = 22/3

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

I(a,b) H(a) Horner multiply before degree2 := mul(22/3, 0)

After

I(a,b) H(a) Horner multiply before degree2 = 0

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

I(a,b) H(a) Horner add degree2 := sum(0, 21/2)

After

I(a,b) H(a) Horner add degree2 = 21/2

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

I(a,b) H(a) Horner multiply before degree1 := mul(21/2, 0)

After

I(a,b) H(a) Horner multiply before degree1 = 0

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

I(a,b) H(a) Horner add degree1 := sum(0, 13)

After

I(a,b) H(a) Horner add degree1 = 13

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

I(a,b) H(a) Horner multiply before degree0 := mul(13, 0)

After

I(a,b) H(a) Horner multiply before degree0 = 0

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

I(a,b) H(a) Horner add degree0 := sum(0, 0)

After

I(a,b) H(a) Horner add degree0 = 0

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

I(a,b) oriented moment := sub(373/12, 0)

After

I(a,b) oriented moment = 373/12

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, S024

S026 · calculation

Before

I(b,c) primitive coefficient1 := div(13, 1)

After

I(b,c) primitive coefficient1 = 13

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, S025

S027 · calculation

Before

I(b,c) primitive coefficient2 := div(21, 2)

After

I(b,c) primitive coefficient2 = 21/2

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, S026

S028 · calculation

Before

I(b,c) primitive coefficient3 := div(22, 3)

After

I(b,c) primitive coefficient3 = 22/3

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, S027

S029 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · implies

Before

[0, 13, 21/2, 22/3, 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 dependencies

Rule: polynomial_primitive_identity · Uses: G001, S029

S031 · implies

Before

I(b,c) H(b) ascending coefficients=[0, 13, 21/2, 22/3, 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 dependencies

Rule: polynomial_evaluation_order · Uses: G001, S030

S032 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

I(b,c) H(b) Horner add degree3 := sum(3/4, 22/3)

After

I(b,c) H(b) Horner add degree3 = 97/12

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

I(b,c) H(b) Horner multiply before degree2 := mul(97/12, 3)

After

I(b,c) H(b) Horner multiply before degree2 = 97/4

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

I(b,c) H(b) Horner add degree2 := sum(97/4, 21/2)

After

I(b,c) H(b) Horner add degree2 = 139/4

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

I(b,c) H(b) Horner multiply before degree1 := mul(139/4, 3)

After

I(b,c) H(b) Horner multiply before degree1 = 417/4

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

I(b,c) H(b) Horner add degree1 := sum(417/4, 13)

After

I(b,c) H(b) Horner add degree1 = 469/4

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

I(b,c) H(b) Horner multiply before degree0 := mul(469/4, 3)

After

I(b,c) H(b) Horner multiply before degree0 = 1407/4

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

I(b,c) H(b) Horner add degree0 := sum(1407/4, 0)

After

I(b,c) H(b) Horner add degree0 = 1407/4

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · implies

Before

I(b,c) H(a) ascending coefficients=[0, 13, 21/2, 22/3, 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 dependencies

Rule: polynomial_evaluation_order · Uses: G001, S039

S041 · calculation

Before

I(b,c) H(a) Horner multiply before degree3 := mul(1/4, 1)

After

I(b,c) H(a) Horner multiply before degree3 = 1/4

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

I(b,c) H(a) Horner add degree3 := sum(1/4, 22/3)

After

I(b,c) H(a) Horner add degree3 = 91/12

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

I(b,c) H(a) Horner multiply before degree2 := mul(91/12, 1)

After

I(b,c) H(a) Horner multiply before degree2 = 91/12

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

I(b,c) H(a) Horner add degree2 := sum(91/12, 21/2)

After

I(b,c) H(a) Horner add degree2 = 217/12

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · calculation

Before

I(b,c) H(a) Horner multiply before degree1 := mul(217/12, 1)

After

I(b,c) H(a) Horner multiply before degree1 = 217/12

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · calculation

Before

I(b,c) H(a) Horner add degree1 := sum(217/12, 13)

After

I(b,c) H(a) Horner add degree1 = 373/12

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · calculation

Before

I(b,c) H(a) Horner multiply before degree0 := mul(373/12, 1)

After

I(b,c) H(a) Horner multiply before degree0 = 373/12

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · calculation

Before

I(b,c) H(a) Horner add degree0 := sum(373/12, 0)

After

I(b,c) H(a) Horner add degree0 = 373/12

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · calculation

Before

I(b,c) oriented moment := sub(1407/4, 373/12)

After

I(b,c) oriented moment = 962/3

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, S048

S050 · calculation

Before

I(a,c) primitive coefficient1 := div(13, 1)

After

I(a,c) primitive coefficient1 = 13

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, S049

S051 · calculation

Before

I(a,c) primitive coefficient2 := div(21, 2)

After

I(a,c) primitive coefficient2 = 21/2

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, S050

S052 · calculation

Before

I(a,c) primitive coefficient3 := div(22, 3)

After

I(a,c) primitive coefficient3 = 22/3

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, S051

S053 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · implies

Before

[0, 13, 21/2, 22/3, 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 dependencies

Rule: polynomial_primitive_identity · Uses: G001, S053

S055 · implies

Before

I(a,c) H(b) ascending coefficients=[0, 13, 21/2, 22/3, 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 dependencies

Rule: polynomial_evaluation_order · Uses: G001, S054

S056 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S055

S057 · calculation

Before

I(a,c) H(b) Horner add degree3 := sum(3/4, 22/3)

After

I(a,c) H(b) Horner add degree3 = 97/12

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S056

S058 · calculation

Before

I(a,c) H(b) Horner multiply before degree2 := mul(97/12, 3)

After

I(a,c) H(b) Horner multiply before degree2 = 97/4

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · calculation

Before

I(a,c) H(b) Horner add degree2 := sum(97/4, 21/2)

After

I(a,c) H(b) Horner add degree2 = 139/4

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S058

S060 · calculation

Before

I(a,c) H(b) Horner multiply before degree1 := mul(139/4, 3)

After

I(a,c) H(b) Horner multiply before degree1 = 417/4

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S059

S061 · calculation

Before

I(a,c) H(b) Horner add degree1 := sum(417/4, 13)

After

I(a,c) H(b) Horner add degree1 = 469/4

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S060

S062 · calculation

Before

I(a,c) H(b) Horner multiply before degree0 := mul(469/4, 3)

After

I(a,c) H(b) Horner multiply before degree0 = 1407/4

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S061

S063 · calculation

Before

I(a,c) H(b) Horner add degree0 := sum(1407/4, 0)

After

I(a,c) H(b) Horner add degree0 = 1407/4

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S062

S064 · implies

Before

I(a,c) H(a) ascending coefficients=[0, 13, 21/2, 22/3, 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 dependencies

Rule: polynomial_evaluation_order · Uses: G001, S063

S065 · calculation

Before

I(a,c) H(a) Horner multiply before degree3 := mul(1/4, 0)

After

I(a,c) H(a) Horner multiply before degree3 = 0

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S064

S066 · calculation

Before

I(a,c) H(a) Horner add degree3 := sum(0, 22/3)

After

I(a,c) H(a) Horner add degree3 = 22/3

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S065

S067 · calculation

Before

I(a,c) H(a) Horner multiply before degree2 := mul(22/3, 0)

After

I(a,c) H(a) Horner multiply before degree2 = 0

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S066

S068 · calculation

Before

I(a,c) H(a) Horner add degree2 := sum(0, 21/2)

After

I(a,c) H(a) Horner add degree2 = 21/2

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S067

S069 · calculation

Before

I(a,c) H(a) Horner multiply before degree1 := mul(21/2, 0)

After

I(a,c) H(a) Horner multiply before degree1 = 0

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S068

S070 · calculation

Before

I(a,c) H(a) Horner add degree1 := sum(0, 13)

After

I(a,c) H(a) Horner add degree1 = 13

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S069

S071 · calculation

Before

I(a,c) H(a) Horner multiply before degree0 := mul(13, 0)

After

I(a,c) H(a) Horner multiply before degree0 = 0

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S070

S072 · calculation

Before

I(a,c) H(a) Horner add degree0 := sum(0, 0)

After

I(a,c) H(a) Horner add degree0 = 0

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S071

S073 · calculation

Before

I(a,c) oriented moment := sub(1407/4, 0)

After

I(a,c) oriented moment = 1407/4

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, S072

S074 · calculation

Before

adjacent oriented moment sum := sum(373/12, 962/3)

After

adjacent oriented moment sum = 1407/4

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, S073

S075 · implies

Before

[H(b)−H(a)]+[H(c)−H(b)]

After

H(c)−H(a)=1407/4

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, 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

(373/12, 962/3, 1407/4)

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: ff31af18647e0a7cd30f5b979985253c0ce7b46e72a93b075c9617cc6a89c542
  • candidate record SHA-256: 1317aa87cb5d5933abd3238214825329327627ee88b7148491e7341f80209d79
  • semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d
  • question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON