Question 32665: Signed Moment Integral of a Monomial

Original question and givens

On all R, f(x)=(15)x^2, 0≤n≤5. Use the exact signed moment c(b^(n+1)−a^(n+1))/(n+1) on oriented endpoints a=-3, b=1. Give a reduced fraction with positive denominator, retaining /1 for integers.

• G001 — On all R, f(x)=(15)x^2, 0≤n≤5. Use the exact signed moment c(b^(n+1)−a^(n+1))/(n+1) on oriented endpoints a=-3, b=1. 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

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

derived = [-27]

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

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication1 = 1

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication2 := mul(1, 1)

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication2 = 1

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication3 := mul(1, 1)

Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication3 = 1

source roles = [Monomial ORIGINAL term upper endpoint, 1, 3]

derived = [1]

Monomial ORIGINAL term ORIGINAL upper power minus lower power := sub(1, -27)

Monomial ORIGINAL term ORIGINAL upper power minus lower power = 28

Monomial ORIGINAL term ORIGINAL coefficient times power difference := mul(15, 28)

Monomial ORIGINAL term ORIGINAL coefficient times power difference = 420

Monomial ORIGINAL term ORIGINAL signed moment over positive exponent := div(420, 3)

Monomial ORIGINAL term ORIGINAL signed moment over positive exponent = 140

source roles = [Monomial ORIGINAL term, 15, 2, -3, 1]

derived = [-27, 1, 28, 140/1]

Reverse checks

• S013 · 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.
• S012 · premise_and_result_audit — Recompute Monomial ORIGINAL term ORIGINAL signed moment over positive exponent with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Monomial ORIGINAL term ORIGINAL coefficient times power difference with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Monomial ORIGINAL term ORIGINAL upper power minus lower power with exact rational arithmetic, not rounded decimals.
• S009 · 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.
• S008 · premise_and_result_audit — Recompute Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication3 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Monomial ORIGINAL term upper endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals.
• S005 · 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.
• 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

140/1

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.