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R0006-03851 · q-20261007-advance-20261009-r06-definite-integral-0051
On all R, f(x)=(0)x+(2). On a=0≤b=2, use N=3 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)=(0)x+(2). On a=0≤b=2, use N=3 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(2, 0)
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, 3)
After
partition step = 2/3
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, 2/3)
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(0, 0)
After
left sample0 x = 0
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
left sample0 mx := mul(0, 0)
After
left sample0 mx = 0
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
left sample0 value := sum(0, 2)
After
left sample0 value = 2
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
left sample0 rectangle := mul(2, 2/3)
After
left sample0 rectangle = 4/3
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
[0, 2, 4/3]
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, 2/3)
After
left sample1 offset = 2/3
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(0, 2/3)
After
left sample1 x = 2/3
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
left sample1 mx := mul(0, 2/3)
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, 2)
After
left sample1 value = 2
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
left sample1 rectangle := mul(2, 2/3)
After
left sample1 rectangle = 4/3
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
[2/3, 2, 4/3]
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, 2/3)
After
left sample2 offset = 4/3
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(0, 4/3)
After
left sample2 x = 4/3
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
left sample2 mx := mul(0, 4/3)
After
left sample2 mx = 0
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
left sample2 value := sum(0, 2)
After
left sample2 value = 2
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
left sample2 rectangle := mul(2, 2/3)
After
left sample2 rectangle = 4/3
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
[4/3, 2, 4/3]
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
finite left sum := sum(4/3, 4/3, 4/3)
After
finite left sum = 4
Conditions: none
Add all N contributions.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
affine integral primitive coefficient1 := div(2, 1)
After
affine integral primitive coefficient1 = 2
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
affine integral primitive coefficient2 := div(0, 2)
After
affine integral primitive coefficient2 = 0
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
[0, 2, 0]
After
H′=the supplied polynomial;integrate from 0 to 2
Conditions: none
Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference.
Rule: polynomial_primitive_identity · Uses: G001, S024
Before
affine integral H(b) ascending coefficients=[0, 2, 0]
After
evaluate at x=2
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, S025
Before
affine integral H(b) Horner multiply before degree0 := mul(2, 2)
After
affine integral H(b) Horner multiply before degree0 = 4
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
affine integral H(b) Horner add degree0 := sum(4, 0)
After
affine integral H(b) Horner add degree0 = 4
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
affine integral H(a) ascending coefficients=[0, 2, 0]
After
evaluate at x=0
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, S028
Before
affine integral H(a) Horner multiply before degree0 := mul(2, 0)
After
affine integral H(a) Horner multiply before degree0 = 0
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
affine integral H(a) Horner add degree0 := sum(0, 0)
After
affine integral H(a) Horner add degree0 = 0
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
affine integral oriented moment := sub(4, 0)
After
affine integral oriented moment = 4
Conditions: none
Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
integral minus left sum := sub(4, 4)
After
integral minus left sum = 0
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, S032
Before
I=4;S_N=4;N=3
After
signed finite-sum difference=0
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, S033
S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S033 · premise_and_result_audit — Recompute integral minus left sum with exact rational arithmetic, not rounded decimals.S032 · premise_and_result_audit — Recompute affine integral oriented moment with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S028 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S024 · premise_and_result_audit — Recompute affine integral primitive coefficient2 with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute affine integral primitive coefficient1 with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute finite left sum 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.0/1
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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