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R0006-03856 · q-20261007-advance-20261009-r06-definite-integral-0056
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.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(5, -1)
After
interval width = 6
Conditions: none
Compute the nonnegative interval width,allowing zero width.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
partition step := div(6, 6)
After
partition step = 1
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)
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(-6, -1)
After
left sample0 mx = 6
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
left sample0 value := sum(6, -7)
After
left sample0 value = -1
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
[-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.
Rule: left_partition_term · Uses: G001, S008
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
left sample1 x := sum(-1, 1)
After
left sample1 x = 0
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
left sample1 mx := mul(-6, 0)
After
left sample1 mx = 0
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
left sample1 value := sum(0, -7)
After
left sample1 value = -7
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
[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.
Rule: left_partition_term · Uses: G001, S014
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
left sample2 x := sum(-1, 2)
After
left sample2 x = 1
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
left sample2 mx := mul(-6, 1)
After
left sample2 mx = -6
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
left sample2 value := sum(-6, -7)
After
left sample2 value = -13
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
[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.
Rule: left_partition_term · Uses: G001, S020
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
left sample3 x := sum(-1, 3)
After
left sample3 x = 2
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
left sample3 mx := mul(-6, 2)
After
left sample3 mx = -12
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
left sample3 value := sum(-12, -7)
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)
After
left sample3 rectangle = -19
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
[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.
Rule: left_partition_term · Uses: G001, S026
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
left sample4 x := sum(-1, 4)
After
left sample4 x = 3
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
left sample4 mx := mul(-6, 3)
After
left sample4 mx = -18
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
left sample4 value := sum(-18, -7)
After
left sample4 value = -25
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
[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.
Rule: left_partition_term · Uses: G001, S032
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
left sample5 x := sum(-1, 5)
After
left sample5 x = 4
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
left sample5 mx := mul(-6, 4)
After
left sample5 mx = -24
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
left sample5 value := sum(-24, -7)
After
left sample5 value = -31
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
[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.
Rule: left_partition_term · Uses: G001, S038
Before
finite left sum := sum(-1, -7, -13, -19, -25, -31)
After
finite left sum = -96
Conditions: none
Add all N contributions.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
[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.
Rule: polynomial_primitive_identity · Uses: G001, S042
Before
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.
Rule: polynomial_evaluation_order · Uses: G001, S043
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S044
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S045
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
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.
Rule: polynomial_evaluation_order · Uses: G001, S048
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S050
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S051
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S054
Before
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.
Rule: finite_sum_vs_integral · Uses: G001, S055
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.-18/1
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.
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