Explore related knowledge and questions →
← Question derivation index · Original question batch
R0006-03858 · q-20261007-advance-20261009-r06-definite-integral-0058
On all R, f(x)=(-8)x+(-5). On a=1≤b=3, use N=8 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)=(-8)x+(-5). On a=1≤b=3, use N=8 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.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.
Before
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.
Rule: polynomial_moment_scope · Uses: G001
Before
interval width := sub(3, 1)
After
interval width = 2
Conditions: none
Compute the nonnegative interval width,allowing zero width.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
partition step := div(2, 8)
After
partition step = 1/4
Conditions: none
Divide by positive integer N to get the exact step.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
left sample0 offset := mul(0, 1/4)
After
left sample0 offset = 0
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
left sample0 x := sum(1, 0)
After
left sample0 x = 1
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
left sample0 mx := mul(-8, 1)
After
left sample0 mx = -8
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
left sample0 value := sum(-8, -5)
After
left sample0 value = -13
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
left sample0 rectangle := mul(-13, 1/4)
After
left sample0 rectangle = -13/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
[1, -13, -13/4]
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.
Rule: left_partition_term · Uses: G001, S008
Before
left sample1 offset := mul(1, 1/4)
After
left sample1 offset = 1/4
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
left sample1 x := sum(1, 1/4)
After
left sample1 x = 5/4
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
left sample1 mx := mul(-8, 5/4)
After
left sample1 mx = -10
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
left sample1 value := sum(-10, -5)
After
left sample1 value = -15
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
left sample1 rectangle := mul(-15, 1/4)
After
left sample1 rectangle = -15/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
[5/4, -15, -15/4]
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.
Rule: left_partition_term · Uses: G001, S014
Before
left sample2 offset := mul(2, 1/4)
After
left sample2 offset = 1/2
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
left sample2 x := sum(1, 1/2)
After
left sample2 x = 3/2
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
left sample2 mx := mul(-8, 3/2)
After
left sample2 mx = -12
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
left sample2 value := sum(-12, -5)
After
left sample2 value = -17
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
left sample2 rectangle := mul(-17, 1/4)
After
left sample2 rectangle = -17/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
[3/2, -17, -17/4]
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.
Rule: left_partition_term · Uses: G001, S020
Before
left sample3 offset := mul(3, 1/4)
After
left sample3 offset = 3/4
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
left sample3 x := sum(1, 3/4)
After
left sample3 x = 7/4
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
left sample3 mx := mul(-8, 7/4)
After
left sample3 mx = -14
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
left sample3 value := sum(-14, -5)
After
left sample3 value = -19
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
left sample3 rectangle := mul(-19, 1/4)
After
left sample3 rectangle = -19/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
[7/4, -19, -19/4]
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.
Rule: left_partition_term · Uses: G001, S026
Before
left sample4 offset := mul(4, 1/4)
After
left sample4 offset = 1
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
left sample4 x := sum(1, 1)
After
left sample4 x = 2
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
left sample4 mx := mul(-8, 2)
After
left sample4 mx = -16
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
left sample4 value := sum(-16, -5)
After
left sample4 value = -21
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
left sample4 rectangle := mul(-21, 1/4)
After
left sample4 rectangle = -21/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
[2, -21, -21/4]
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.
Rule: left_partition_term · Uses: G001, S032
Before
left sample5 offset := mul(5, 1/4)
After
left sample5 offset = 5/4
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
left sample5 x := sum(1, 5/4)
After
left sample5 x = 9/4
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
left sample5 mx := mul(-8, 9/4)
After
left sample5 mx = -18
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
left sample5 value := sum(-18, -5)
After
left sample5 value = -23
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
left sample5 rectangle := mul(-23, 1/4)
After
left sample5 rectangle = -23/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
[9/4, -23, -23/4]
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.
Rule: left_partition_term · Uses: G001, S038
Before
left sample6 offset := mul(6, 1/4)
After
left sample6 offset = 3/2
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
left sample6 x := sum(1, 3/2)
After
left sample6 x = 5/2
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
left sample6 mx := mul(-8, 5/2)
After
left sample6 mx = -20
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
left sample6 value := sum(-20, -5)
After
left sample6 value = -25
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
left sample6 rectangle := mul(-25, 1/4)
After
left sample6 rectangle = -25/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
[5/2, -25, -25/4]
After
finite left-sum term index=6
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S044
Before
left sample7 offset := mul(7, 1/4)
After
left sample7 offset = 7/4
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S045
Before
left sample7 x := sum(1, 7/4)
After
left sample7 x = 11/4
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
left sample7 mx := mul(-8, 11/4)
After
left sample7 mx = -22
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
left sample7 value := sum(-22, -5)
After
left sample7 value = -27
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
left sample7 rectangle := mul(-27, 1/4)
After
left sample7 rectangle = -27/4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
[11/4, -27, -27/4]
After
finite left-sum term index=7
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S050
Before
finite left sum := sum(-13/4, -15/4, -17/4, -19/4, -21/4, -23/4, -25/4, -27/4)
After
finite left sum = -40
Conditions: none
Add all N contributions.
Rule: exact_rational_arithmetic · Uses: G001, S051
Before
affine integral primitive coefficient1 := div(-5, 1)
After
affine integral primitive coefficient1 = -5
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
affine integral primitive coefficient2 := div(-8, 2)
After
affine integral primitive coefficient2 = -4
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
[0, -5, -4]
After
H′=the supplied polynomial;integrate from 1 to 3
Conditions: none
Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference.
Rule: polynomial_primitive_identity · Uses: G001, S054
Before
affine integral H(b) ascending coefficients=[0, -5, -4]
After
evaluate at x=3
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.
Rule: polynomial_evaluation_order · Uses: G001, S055
Before
affine integral H(b) Horner multiply before degree1 := mul(-4, 3)
After
affine integral H(b) Horner multiply before degree1 = -12
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S056
Before
affine integral H(b) Horner add degree1 := sum(-12, -5)
After
affine integral H(b) Horner add degree1 = -17
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S057
Before
affine integral H(b) Horner multiply before degree0 := mul(-17, 3)
After
affine integral H(b) Horner multiply before degree0 = -51
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
affine integral H(b) Horner add degree0 := sum(-51, 0)
After
affine integral H(b) Horner add degree0 = -51
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S059
Before
affine integral H(a) ascending coefficients=[0, -5, -4]
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.
Rule: polynomial_evaluation_order · Uses: G001, S060
Before
affine integral H(a) Horner multiply before degree1 := mul(-4, 1)
After
affine integral H(a) Horner multiply before degree1 = -4
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S061
Before
affine integral H(a) Horner add degree1 := sum(-4, -5)
After
affine integral H(a) Horner add degree1 = -9
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S062
Before
affine integral H(a) Horner multiply before degree0 := mul(-9, 1)
After
affine integral H(a) Horner multiply before degree0 = -9
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S063
Before
affine integral H(a) Horner add degree0 := sum(-9, 0)
After
affine integral H(a) Horner add degree0 = -9
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S064
Before
affine integral oriented moment := sub(-51, -9)
After
affine integral oriented moment = -42
Conditions: none
Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign.
Rule: exact_rational_arithmetic · Uses: G001, S065
Before
integral minus left sum := sub(-42, -40)
After
integral minus left sum = -2
Conditions: none
The requested direction is I−S_N;a negative slope may give a negative difference,not absolute error.
Rule: exact_rational_arithmetic · Uses: G001, S066
Before
I=-42;S_N=-40;N=8
After
signed finite-sum difference=-2
Conditions: none
This is the exact difference for finite N,not evidence of a general convergence rate or integrability of arbitrary functions.
Rule: finite_sum_vs_integral · Uses: G001, S067
S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S067 · premise_and_result_audit — Recompute integral minus left sum with exact rational arithmetic, not rounded decimals.S066 · premise_and_result_audit — Recompute affine integral oriented moment with exact rational arithmetic, not rounded decimals.S065 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.S064 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S063 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals.S062 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S060 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.S059 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S058 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals.S057 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S054 · premise_and_result_audit — Recompute affine integral primitive coefficient2 with exact rational arithmetic, not rounded decimals.S053 · premise_and_result_audit — Recompute affine integral primitive coefficient1 with exact rational arithmetic, not rounded decimals.S052 · premise_and_result_audit — Recompute finite left sum with exact rational arithmetic, not rounded decimals.S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S050 · premise_and_result_audit — Recompute left sample7 rectangle with exact rational arithmetic, not rounded decimals.S049 · premise_and_result_audit — Recompute left sample7 value with exact rational arithmetic, not rounded decimals.S048 · premise_and_result_audit — Recompute left sample7 mx with exact rational arithmetic, not rounded decimals.S047 · premise_and_result_audit — Recompute left sample7 x with exact rational arithmetic, not rounded decimals.S046 · premise_and_result_audit — Recompute left sample7 offset with exact rational arithmetic, not rounded decimals.S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S044 · premise_and_result_audit — Recompute left sample6 rectangle with exact rational arithmetic, not rounded decimals.S043 · premise_and_result_audit — Recompute left sample6 value with exact rational arithmetic, not rounded decimals.S042 · premise_and_result_audit — Recompute left sample6 mx with exact rational arithmetic, not rounded decimals.S041 · premise_and_result_audit — Recompute left sample6 x with exact rational arithmetic, not rounded decimals.S040 · premise_and_result_audit — Recompute left sample6 offset 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.-2/1
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: ed7fbc4aa0a41d1c119bff38d0282b23d8611423c8f8d7a88c8f86bc98a03a15candidate record SHA-256: 0117ae2c0994177884bdb12c866f5f0baeb98b9ed84caff0355fcc450087577fsemantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED