# Question 3856: Exact Moment Integral Minus a Finite Left Sum [Return to catalogue page 20](https://ocme.evemisslab.com/en/questions/page/20/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261009-r06-definite-integral-foundations/#q-20261007-advance-20261009-r06-definite-integral-0056) 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. R0006-03856 · q-20261007-advance-20261009-r06-definite-integral-0056 ## Original question and givens On all R, f(x)=(-6)x+(-7). On a=-1≤b=5, use N=6 uniform left cells. Find the signed error I−S_N, exact moment integral minus the finite left sum. Use reduced positive-denominator fractions, retaining /1 for integers. - G001 — On all R, f(x)=(-6)x+(-7). On a=-1≤b=5, use N=6 uniform left cells. Find the signed error I−S_N, exact moment integral minus the finite left sum. 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. - [mko-definite-integral](https://ocme.evemisslab.com/en/mko/mko-definite-integral/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: polynomial_moment_scope · Uses: G001 S002 · calculationBefore interval width := sub(5, -1) After interval width = 6 Conditions: none Compute the nonnegative interval width,allowing zero width. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore partition step := div(6, 6) After partition step = 1 Conditions: none Divide by positive integer N to get the exact step. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore left sample0 offset := mul(0, 1) After left sample0 offset = 0 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore left sample0 x := sum(-1, 0) After left sample0 x = -1 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore left sample0 mx := mul(-6, -1) After left sample0 mx = 6 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore left sample0 value := sum(6, -7) After left sample0 value = -1 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore left sample0 rectangle := mul(-1, 1) After left sample0 rectangle = -1 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore [-1, -1, -1] After finite left-sum term index=0 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S008 S010 · calculationBefore left sample1 offset := mul(1, 1) After left sample1 offset = 1 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore left sample1 x := sum(-1, 1) After left sample1 x = 0 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore left sample1 mx := mul(-6, 0) After left sample1 mx = 0 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore left sample1 value := sum(0, -7) After left sample1 value = -7 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore left sample1 rectangle := mul(-7, 1) After left sample1 rectangle = -7 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore [0, -7, -7] After finite left-sum term index=1 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S014 S016 · calculationBefore left sample2 offset := mul(2, 1) After left sample2 offset = 2 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore left sample2 x := sum(-1, 2) After left sample2 x = 1 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore left sample2 mx := mul(-6, 1) After left sample2 mx = -6 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore left sample2 value := sum(-6, -7) After left sample2 value = -13 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore left sample2 rectangle := mul(-13, 1) After left sample2 rectangle = -13 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · impliesBefore [1, -13, -13] After finite left-sum term index=2 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S020 S022 · calculationBefore left sample3 offset := mul(3, 1) After left sample3 offset = 3 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore left sample3 x := sum(-1, 3) After left sample3 x = 2 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore left sample3 mx := mul(-6, 2) After left sample3 mx = -12 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore left sample3 value := sum(-12, -7) After left sample3 value = -19 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore left sample3 rectangle := mul(-19, 1) After left sample3 rectangle = -19 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · impliesBefore [2, -19, -19] After finite left-sum term index=3 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S026 S028 · calculationBefore left sample4 offset := mul(4, 1) After left sample4 offset = 4 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore left sample4 x := sum(-1, 4) After left sample4 x = 3 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore left sample4 mx := mul(-6, 3) After left sample4 mx = -18 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore left sample4 value := sum(-18, -7) After left sample4 value = -25 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore left sample4 rectangle := mul(-25, 1) After left sample4 rectangle = -25 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · impliesBefore [3, -25, -25] After finite left-sum term index=4 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S032 S034 · calculationBefore left sample5 offset := mul(5, 1) After left sample5 offset = 5 Conditions: none Left samples use i=0,...,N−1,not the final right endpoint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore left sample5 x := sum(-1, 5) After left sample5 x = 4 Conditions: none Add the indexed offset to the start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore left sample5 mx := mul(-6, 4) After left sample5 mx = -24 Conditions: none Evaluate the product term of the affine integrand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore left sample5 value := sum(-24, -7) After left sample5 value = -31 Conditions: none Add c to obtain this sample value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore left sample5 rectangle := mul(-31, 1) After left sample5 rectangle = -31 Conditions: none Multiply by the step to get the signed rectangle contribution,without absolute values. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · impliesBefore [4, -31, -31] After finite left-sum term index=5 Conditions: none List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof. Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S038 S040 · calculationBefore finite left sum := sum(-1, -7, -13, -19, -25, -31) After finite left sum = -96 Conditions: none Add all N contributions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore affine integral primitive coefficient1 := div(-7, 1) After affine integral primitive coefficient1 = -7 Conditions: none The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore affine integral primitive coefficient2 := div(-6, 2) After affine integral primitive coefficient2 = -3 Conditions: none The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · impliesBefore [0, -7, -3] After H′=the supplied polynomial;integrate from -1 to 5 Conditions: none Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference. Step identifiers and dependenciesRule: polynomial_primitive_identity · Uses: G001, S042 S044 · impliesBefore affine integral H(b) ascending coefficients=[0, -7, -3] After evaluate at x=5 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 dependenciesRule: polynomial_evaluation_order · Uses: G001, S043 S045 · calculationBefore affine integral H(b) Horner multiply before degree1 := mul(-3, 5) After affine integral H(b) Horner multiply before degree1 = -15 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore affine integral H(b) Horner add degree1 := sum(-15, -7) After affine integral H(b) Horner add degree1 = -22 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore affine integral H(b) Horner multiply before degree0 := mul(-22, 5) After affine integral H(b) Horner multiply before degree0 = -110 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · calculationBefore affine integral H(b) Horner add degree0 := sum(-110, 0) After affine integral H(b) Horner add degree0 = -110 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · impliesBefore affine integral H(a) ascending coefficients=[0, -7, -3] 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 dependenciesRule: polynomial_evaluation_order · Uses: G001, S048 S050 · calculationBefore affine integral H(a) Horner multiply before degree1 := mul(-3, -1) After affine integral H(a) Horner multiply before degree1 = 3 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · calculationBefore affine integral H(a) Horner add degree1 := sum(3, -7) After affine integral H(a) Horner add degree1 = -4 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S050 S052 · calculationBefore affine integral H(a) Horner multiply before degree0 := mul(-4, -1) After affine integral H(a) Horner multiply before degree0 = 4 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S051 S053 · calculationBefore affine integral H(a) Horner add degree0 := sum(4, 0) After affine integral H(a) Horner add degree0 = 4 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052 S054 · calculationBefore affine integral oriented moment := sub(-110, 4) After affine integral oriented moment = -114 Conditions: none Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S053 S055 · calculationBefore integral minus left sum := sub(-114, -96) After integral minus left sum = -18 Conditions: none The requested direction is I−S_N;a negative slope may give a negative difference,not absolute error. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S054 S056 · impliesBefore I=-114;S_N=-96;N=6 After signed finite-sum difference=-18 Conditions: none This is the exact difference for finite N,not evidence of a general convergence rate or integrability of arbitrary functions. Step identifiers and dependenciesRule: finite_sum_vs_integral · Uses: G001, S055 ## Reverse checks - S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S055 · premise_and_result_audit — Recompute integral minus left sum with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Recompute affine integral oriented moment with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S048 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S042 · premise_and_result_audit — Recompute affine integral primitive coefficient2 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute affine integral primitive coefficient1 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute finite left sum with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S038 · premise_and_result_audit — Recompute left sample5 rectangle with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute left sample5 value with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute left sample5 mx with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute left sample5 x with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute left sample5 offset with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S032 · premise_and_result_audit — Recompute left sample4 rectangle with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute left sample4 value with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute left sample4 mx with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute left sample4 x with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute left sample4 offset with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Recompute left sample3 rectangle with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute left sample3 value with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute left sample3 mx with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute left sample3 x with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute left sample3 offset with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Recompute left sample2 rectangle with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute left sample2 value with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute left sample2 mx with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute left sample2 x with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute left sample2 offset with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Recompute left sample1 rectangle with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute left sample1 value with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute left sample1 mx with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute left sample1 x with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute left sample1 offset with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute left sample0 rectangle with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute left sample0 value with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute left sample0 mx with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute left sample0 x with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute left sample0 offset with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute partition step with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute interval width 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 -18/1 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 6ccae1ccfc18c85c1ed974b6559d89a6973c86af5454a6381ca4baa286079fa4 - candidate record SHA-256: c4314b7b792b64af97427c354cc8771e2bafabfd01371ea8edaf03b9291f7c9e - semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d - question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261009-r06-definite-integral-0056.json)