# Question 28145: Each Trapezoid Uses Its Own Width and Endpoint Average [Return to catalogue page 141](https://ocme.evemisslab.com/en/questions/page/141/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations/#q-20261015-pulse-20261119-r02-0045) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0055-28145 · q-20261015-pulse-20261119-r02-0045 ## Original question and givens Use strictly increasing nodes=[0, 1, 22], aligned endpoint values=[19, 20, 22]. Give (nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention). - G001 — Use strictly increasing nodes=[0, 1, 22], aligned endpoint values=[19, 20, 22]. 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](https://ocme.evemisslab.com/en/mko/mko-trapezoidal-quadrature/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-mod50-trap-composite-value, trap-composite-value] After derived = [[nodes, endpoint_values, individual_trapezoid_terms, trapezoid_value, width_convention]] Conditions: none List this scalar/label/set/point answer role or mode-specific ordered fields,not an imposed universal tuple.Retain legacy domain/underscore operation and integer fractions with/1;original bilingual snapshots stay unchanged. Step identifiers and dependenciesRule: source_answer_roles_or_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore Trapezoid ORIGINAL ORIGINAL width0 := sub(1, 0) After Trapezoid ORIGINAL ORIGINAL width0 = 1 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Trapezoid ORIGINAL ORIGINAL width1 := sub(22, 1) After Trapezoid ORIGINAL ORIGINAL width1 = 21 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Trapezoid ORIGINAL ORIGINAL width over2 interval0 := div(1, 2) After Trapezoid ORIGINAL ORIGINAL width over2 interval0 = 1/2 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Trapezoid ORIGINAL ORIGINAL endpoint sum interval0 := sum(19, 20) After Trapezoid ORIGINAL ORIGINAL endpoint sum interval0 = 39 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Trapezoid ORIGINAL ORIGINAL individual trapezoid interval0 := mul(1/2, 39) After Trapezoid ORIGINAL ORIGINAL individual trapezoid interval0 = 39/2 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [Trapezoid ORIGINAL, 0, 0, 1, 19, 20] After derived = [1, 1/2, 39, 39/2] Conditions: none Use the own width and aligned endpoints of this ORIGINAL interval times the endpoint average.Widths may differ and values may be negative;do not sort or replace all widths with the first. Step identifiers and dependenciesRule: Trapezoid_EACH_original_interval_own_width_and_aligned_endpoints · Uses: G001, S006 S008 · calculationBefore Trapezoid ORIGINAL ORIGINAL width over2 interval1 := div(21, 2) After Trapezoid ORIGINAL ORIGINAL width over2 interval1 = 21/2 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Trapezoid ORIGINAL ORIGINAL endpoint sum interval1 := sum(20, 22) After Trapezoid ORIGINAL ORIGINAL endpoint sum interval1 = 42 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Trapezoid ORIGINAL ORIGINAL individual trapezoid interval1 := mul(21/2, 42) After Trapezoid ORIGINAL ORIGINAL individual trapezoid interval1 = 441 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore source roles = [Trapezoid ORIGINAL, 1, 1, 22, 20, 22] After derived = [21, 21/2, 42, 441] Conditions: none Use the own width and aligned endpoints of this ORIGINAL interval times the endpoint average.Widths may differ and values may be negative;do not sort or replace all widths with the first. Step identifiers and dependenciesRule: Trapezoid_EACH_original_interval_own_width_and_aligned_endpoints · Uses: G001, S010 S012 · calculationBefore Trapezoid ORIGINAL ORIGINAL trapezoid running total0 := sum(0, 39/2) After Trapezoid ORIGINAL ORIGINAL trapezoid running total0 = 39/2 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Trapezoid ORIGINAL ORIGINAL trapezoid running total1 := sum(39/2, 441) After Trapezoid ORIGINAL ORIGINAL trapezoid running total1 = 921/2 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore source roles = [Trapezoid ORIGINAL, [0, 1, 22], [19, 20, 22]] After derived = [[0, 1, 22], [19, 20, 22], [1, 21], [39/2, 441], 921/2] Conditions: none A complete partition is STRICTLY increasing in original order,with n+1 aligned nodes/values for n intervals;the24-node cap is not the vector8 cap.The table determines this sum,not a unique integral without a full integrand. Step identifiers and dependenciesRule: Trapezoid_COMPLETE_partition_n_intervals_nplus1_nodes_no_unique_integrand · Uses: G001, S013 ## 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, 22], [19, 20, 22], [39/2, 441], 921/2, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: ce3e7144f25d71a3b8ae7cdd609014780b06717f934be689c12a2b434b3994b5 - candidate record SHA-256: 1b4cce4b867767b351cac808a25f4c4fe61bddda1c35955a56f4d864661a8333 - semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683 - question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-pulse-20261119-r02-0045.json)