Question 28128: Each Trapezoid Uses Its Own Width and Endpoint Average

Original question and givens

Use strictly increasing nodes=[0, 1, 5], aligned endpoint values=[2, 3, 5]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).

• G001 — Use strictly increasing nodes=[0, 1, 5], aligned endpoint values=[2, 3, 5]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention).

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-trapezoidal-quadrature

Step-by-step derivation

source roles = [qf-mod50-trap-composite-value, trap-composite-value]

derived = [[nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention]]

Trapezoid ORIGINAL ORIGINAL width0 := sub(1, 0)

Trapezoid ORIGINAL ORIGINAL width0 = 1

Trapezoid ORIGINAL ORIGINAL width1 := sub(5, 1)

Trapezoid ORIGINAL ORIGINAL width1 = 4

Trapezoid ORIGINAL ORIGINAL width over2 interval0 := div(1, 2)

Trapezoid ORIGINAL ORIGINAL width over2 interval0 = 1/2

Trapezoid ORIGINAL ORIGINAL endpoint sum interval0 := sum(2, 3)

Trapezoid ORIGINAL ORIGINAL endpoint sum interval0 = 5

Trapezoid ORIGINAL ORIGINAL individual trapezoid interval0 := mul(1/2, 5)

Trapezoid ORIGINAL ORIGINAL individual trapezoid interval0 = 5/2

source roles = [Trapezoid ORIGINAL, 0, 0, 1, 2, 3]

derived = [1, 1/2, 5, 5/2]

Trapezoid ORIGINAL ORIGINAL width over2 interval1 := div(4, 2)

Trapezoid ORIGINAL ORIGINAL width over2 interval1 = 2

Trapezoid ORIGINAL ORIGINAL endpoint sum interval1 := sum(3, 5)

Trapezoid ORIGINAL ORIGINAL endpoint sum interval1 = 8

Trapezoid ORIGINAL ORIGINAL individual trapezoid interval1 := mul(2, 8)

Trapezoid ORIGINAL ORIGINAL individual trapezoid interval1 = 16

source roles = [Trapezoid ORIGINAL, 1, 1, 5, 3, 5]

derived = [4, 2, 8, 16]

Trapezoid ORIGINAL ORIGINAL trapezoid running total0 := sum(0, 5/2)

Trapezoid ORIGINAL ORIGINAL trapezoid running total0 = 5/2

Trapezoid ORIGINAL ORIGINAL trapezoid running total1 := sum(5/2, 16)

Trapezoid ORIGINAL ORIGINAL trapezoid running total1 = 37/2

source roles = [Trapezoid ORIGINAL, [0, 1, 5], [2, 3, 5]]

derived = [[0, 1, 5], [2, 3, 5], [1, 4], [5/2, 16], 37/2]

Reverse checks

• S014 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S013 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL trapezoid running total1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL trapezoid running total0 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S010 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL individual trapezoid interval1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL endpoint sum interval1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL width over2 interval1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S006 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL individual trapezoid interval0 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL endpoint sum interval0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL width over2 interval0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL width1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Trapezoid ORIGINAL ORIGINAL width0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.

Result

([0, 1, 5], [2, 3, 5], [5/2, 16], 37/2, each_subinterval_uses_its_own_width)

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.