Question 32674: Signed Moment Integral of a Monomial

Original question and givens

On all R, f(x)=(-24)x^5, 0≤n≤5. Use the exact signed moment c(b^(n+1)−a^(n+1))/(n+1) on oriented endpoints a=-3, b=2. Give a reduced fraction with positive denominator, retaining /1 for integers.

• G001 — On all R, f(x)=(-24)x^5, 0≤n≤5. Use the exact signed moment c(b^(n+1)−a^(n+1))/(n+1) on oriented endpoints a=-3, b=2. Give a reduced fraction with positive denominator, 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

source roles = [qf-integral-monomial, monomial_integral, reals, []]

derived = [[signed_moment_fraction_RETAIN1]]

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication1 := mul(1, -3)

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication1 = -3

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication2 := mul(-3, -3)

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication2 = 9

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication3 := mul(9, -3)

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication3 = -27

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication4 := mul(-27, -3)

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication4 = 81

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication5 := mul(81, -3)

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication5 = -243

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication6 := mul(-243, -3)

Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication6 = 729

source roles = [Monomial ORIGINAL term lower endpoint, -3, 6]

derived = [729]

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication1 := mul(1, 2)

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication1 = 2

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication2 := mul(2, 2)

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication2 = 4

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication3 := mul(4, 2)

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication3 = 8

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication4 := mul(8, 2)

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication4 = 16

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication5 := mul(16, 2)

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication5 = 32

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication6 := mul(32, 2)

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication6 = 64

source roles = [Monomial ORIGINAL term upper endpoint, 2, 6]

derived = [64]

Monomial ORIGINAL term ORIGINAL upper power minus lower power := sub(64, 729)

Monomial ORIGINAL term ORIGINAL upper power minus lower power = -665

Monomial ORIGINAL term ORIGINAL coefficient times power difference := mul(-24, -665)

Monomial ORIGINAL term ORIGINAL coefficient times power difference = 15960

Monomial ORIGINAL term ORIGINAL signed moment over positive exponent := div(15960, 6)

Monomial ORIGINAL term ORIGINAL signed moment over positive exponent = 2660

source roles = [Monomial ORIGINAL term, -24, 5, -3, 2]

derived = [729, 64, -665, 2660/1]

Reverse checks

• S019 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S018 · premise_and_result_audit — Recompute Monomial ORIGINAL term ORIGINAL signed moment over positive exponent with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Monomial ORIGINAL term ORIGINAL coefficient times power difference with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Monomial ORIGINAL term ORIGINAL upper power minus lower power with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S014 · premise_and_result_audit — Recompute Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication6 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication5 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication4 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication3 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S007 · premise_and_result_audit — Recompute Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication6 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication5 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication4 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication3 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Monomial ORIGINAL term lower endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

2660/1

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.