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← Question derivation index · Original question batch
R0006-03871 · q-20261007-advance-20261009-r06-definite-integral-0071
On all R, f(x)=(21)x+(8). On a=-2≤b=1, use N=21 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)=(21)x+(8). On a=-2≤b=1, use N=21 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(1, -2)
After
interval width = 3
Conditions: none
Compute the nonnegative interval width,allowing zero width.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
partition step := div(3, 21)
After
partition step = 1/7
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, 1/7)
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(-2, 0)
After
left sample0 x = -2
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
left sample0 mx := mul(21, -2)
After
left sample0 mx = -42
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
left sample0 value := sum(-42, 8)
After
left sample0 value = -34
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
left sample0 rectangle := mul(-34, 1/7)
After
left sample0 rectangle = -34/7
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
[-2, -34, -34/7]
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, 1/7)
After
left sample1 offset = 1/7
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(-2, 1/7)
After
left sample1 x = -13/7
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
left sample1 mx := mul(21, -13/7)
After
left sample1 mx = -39
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
left sample1 value := sum(-39, 8)
After
left sample1 value = -31
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
left sample1 rectangle := mul(-31, 1/7)
After
left sample1 rectangle = -31/7
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
[-13/7, -31, -31/7]
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, 1/7)
After
left sample2 offset = 2/7
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(-2, 2/7)
After
left sample2 x = -12/7
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
left sample2 mx := mul(21, -12/7)
After
left sample2 mx = -36
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
left sample2 value := sum(-36, 8)
After
left sample2 value = -28
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
left sample2 rectangle := mul(-28, 1/7)
After
left sample2 rectangle = -4
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
[-12/7, -28, -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, 1/7)
After
left sample3 offset = 3/7
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(-2, 3/7)
After
left sample3 x = -11/7
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
left sample3 mx := mul(21, -11/7)
After
left sample3 mx = -33
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
left sample3 value := sum(-33, 8)
After
left sample3 value = -25
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
left sample3 rectangle := mul(-25, 1/7)
After
left sample3 rectangle = -25/7
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
[-11/7, -25, -25/7]
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, 1/7)
After
left sample4 offset = 4/7
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(-2, 4/7)
After
left sample4 x = -10/7
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
left sample4 mx := mul(21, -10/7)
After
left sample4 mx = -30
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
left sample4 value := sum(-30, 8)
After
left sample4 value = -22
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
left sample4 rectangle := mul(-22, 1/7)
After
left sample4 rectangle = -22/7
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
[-10/7, -22, -22/7]
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, 1/7)
After
left sample5 offset = 5/7
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(-2, 5/7)
After
left sample5 x = -9/7
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
left sample5 mx := mul(21, -9/7)
After
left sample5 mx = -27
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
left sample5 value := sum(-27, 8)
After
left sample5 value = -19
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
left sample5 rectangle := mul(-19, 1/7)
After
left sample5 rectangle = -19/7
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
[-9/7, -19, -19/7]
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, 1/7)
After
left sample6 offset = 6/7
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(-2, 6/7)
After
left sample6 x = -8/7
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
left sample6 mx := mul(21, -8/7)
After
left sample6 mx = -24
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
left sample6 value := sum(-24, 8)
After
left sample6 value = -16
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
left sample6 rectangle := mul(-16, 1/7)
After
left sample6 rectangle = -16/7
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
[-8/7, -16, -16/7]
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, 1/7)
After
left sample7 offset = 1
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(-2, 1)
After
left sample7 x = -1
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
left sample7 mx := mul(21, -1)
After
left sample7 mx = -21
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
left sample7 value := sum(-21, 8)
After
left sample7 value = -13
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
left sample7 rectangle := mul(-13, 1/7)
After
left sample7 rectangle = -13/7
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
[-1, -13, -13/7]
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
left sample8 offset := mul(8, 1/7)
After
left sample8 offset = 8/7
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S051
Before
left sample8 x := sum(-2, 8/7)
After
left sample8 x = -6/7
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
left sample8 mx := mul(21, -6/7)
After
left sample8 mx = -18
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
left sample8 value := sum(-18, 8)
After
left sample8 value = -10
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S054
Before
left sample8 rectangle := mul(-10, 1/7)
After
left sample8 rectangle = -10/7
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
[-6/7, -10, -10/7]
After
finite left-sum term index=8
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, S056
Before
left sample9 offset := mul(9, 1/7)
After
left sample9 offset = 9/7
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S057
Before
left sample9 x := sum(-2, 9/7)
After
left sample9 x = -5/7
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
left sample9 mx := mul(21, -5/7)
After
left sample9 mx = -15
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S059
Before
left sample9 value := sum(-15, 8)
After
left sample9 value = -7
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S060
Before
left sample9 rectangle := mul(-7, 1/7)
After
left sample9 rectangle = -1
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S061
Before
[-5/7, -7, -1]
After
finite left-sum term index=9
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, S062
Before
left sample10 offset := mul(10, 1/7)
After
left sample10 offset = 10/7
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S063
Before
left sample10 x := sum(-2, 10/7)
After
left sample10 x = -4/7
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S064
Before
left sample10 mx := mul(21, -4/7)
After
left sample10 mx = -12
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S065
Before
left sample10 value := sum(-12, 8)
After
left sample10 value = -4
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S066
Before
left sample10 rectangle := mul(-4, 1/7)
After
left sample10 rectangle = -4/7
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S067
Before
[-4/7, -4, -4/7]
After
finite left-sum term index=10
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, S068
Before
left sample11 offset := mul(11, 1/7)
After
left sample11 offset = 11/7
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S069
Before
left sample11 x := sum(-2, 11/7)
After
left sample11 x = -3/7
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S070
Before
left sample11 mx := mul(21, -3/7)
After
left sample11 mx = -9
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S071
Before
left sample11 value := sum(-9, 8)
After
left sample11 value = -1
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S072
Before
left sample11 rectangle := mul(-1, 1/7)
After
left sample11 rectangle = -1/7
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S073
Before
[-3/7, -1, -1/7]
After
finite left-sum term index=11
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, S074
Before
left sample12 offset := mul(12, 1/7)
After
left sample12 offset = 12/7
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S075
Before
left sample12 x := sum(-2, 12/7)
After
left sample12 x = -2/7
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S076
Before
left sample12 mx := mul(21, -2/7)
After
left sample12 mx = -6
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S077
Before
left sample12 value := sum(-6, 8)
After
left sample12 value = 2
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S078
Before
left sample12 rectangle := mul(2, 1/7)
After
left sample12 rectangle = 2/7
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S079
Before
[-2/7, 2, 2/7]
After
finite left-sum term index=12
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, S080
Before
left sample13 offset := mul(13, 1/7)
After
left sample13 offset = 13/7
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S081
Before
left sample13 x := sum(-2, 13/7)
After
left sample13 x = -1/7
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S082
Before
left sample13 mx := mul(21, -1/7)
After
left sample13 mx = -3
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S083
Before
left sample13 value := sum(-3, 8)
After
left sample13 value = 5
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S084
Before
left sample13 rectangle := mul(5, 1/7)
After
left sample13 rectangle = 5/7
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S085
Before
[-1/7, 5, 5/7]
After
finite left-sum term index=13
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, S086
Before
left sample14 offset := mul(14, 1/7)
After
left sample14 offset = 2
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S087
Before
left sample14 x := sum(-2, 2)
After
left sample14 x = 0
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S088
Before
left sample14 mx := mul(21, 0)
After
left sample14 mx = 0
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S089
Before
left sample14 value := sum(0, 8)
After
left sample14 value = 8
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S090
Before
left sample14 rectangle := mul(8, 1/7)
After
left sample14 rectangle = 8/7
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S091
Before
[0, 8, 8/7]
After
finite left-sum term index=14
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, S092
Before
left sample15 offset := mul(15, 1/7)
After
left sample15 offset = 15/7
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S093
Before
left sample15 x := sum(-2, 15/7)
After
left sample15 x = 1/7
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S094
Before
left sample15 mx := mul(21, 1/7)
After
left sample15 mx = 3
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S095
Before
left sample15 value := sum(3, 8)
After
left sample15 value = 11
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S096
Before
left sample15 rectangle := mul(11, 1/7)
After
left sample15 rectangle = 11/7
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S097
Before
[1/7, 11, 11/7]
After
finite left-sum term index=15
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, S098
Before
left sample16 offset := mul(16, 1/7)
After
left sample16 offset = 16/7
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S099
Before
left sample16 x := sum(-2, 16/7)
After
left sample16 x = 2/7
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S100
Before
left sample16 mx := mul(21, 2/7)
After
left sample16 mx = 6
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S101
Before
left sample16 value := sum(6, 8)
After
left sample16 value = 14
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S102
Before
left sample16 rectangle := mul(14, 1/7)
After
left sample16 rectangle = 2
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S103
Before
[2/7, 14, 2]
After
finite left-sum term index=16
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, S104
Before
left sample17 offset := mul(17, 1/7)
After
left sample17 offset = 17/7
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S105
Before
left sample17 x := sum(-2, 17/7)
After
left sample17 x = 3/7
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S106
Before
left sample17 mx := mul(21, 3/7)
After
left sample17 mx = 9
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S107
Before
left sample17 value := sum(9, 8)
After
left sample17 value = 17
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S108
Before
left sample17 rectangle := mul(17, 1/7)
After
left sample17 rectangle = 17/7
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S109
Before
[3/7, 17, 17/7]
After
finite left-sum term index=17
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, S110
Before
left sample18 offset := mul(18, 1/7)
After
left sample18 offset = 18/7
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S111
Before
left sample18 x := sum(-2, 18/7)
After
left sample18 x = 4/7
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S112
Before
left sample18 mx := mul(21, 4/7)
After
left sample18 mx = 12
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S113
Before
left sample18 value := sum(12, 8)
After
left sample18 value = 20
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S114
Before
left sample18 rectangle := mul(20, 1/7)
After
left sample18 rectangle = 20/7
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S115
Before
[4/7, 20, 20/7]
After
finite left-sum term index=18
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, S116
Before
left sample19 offset := mul(19, 1/7)
After
left sample19 offset = 19/7
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S117
Before
left sample19 x := sum(-2, 19/7)
After
left sample19 x = 5/7
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S118
Before
left sample19 mx := mul(21, 5/7)
After
left sample19 mx = 15
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S119
Before
left sample19 value := sum(15, 8)
After
left sample19 value = 23
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S120
Before
left sample19 rectangle := mul(23, 1/7)
After
left sample19 rectangle = 23/7
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S121
Before
[5/7, 23, 23/7]
After
finite left-sum term index=19
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, S122
Before
left sample20 offset := mul(20, 1/7)
After
left sample20 offset = 20/7
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S123
Before
left sample20 x := sum(-2, 20/7)
After
left sample20 x = 6/7
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S124
Before
left sample20 mx := mul(21, 6/7)
After
left sample20 mx = 18
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S125
Before
left sample20 value := sum(18, 8)
After
left sample20 value = 26
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S126
Before
left sample20 rectangle := mul(26, 1/7)
After
left sample20 rectangle = 26/7
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S127
Before
[6/7, 26, 26/7]
After
finite left-sum term index=20
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, S128
Before
finite left sum := sum(-34/7, -31/7, -4, -25/7, -22/7, -19/7, -16/7, -13/7, -10/7, -1, -4/7, -1/7, 2/7, 5/7, 8/7, 11/7, 2, 17/7, 20/7, 23/7, 26/7)
After
finite left sum = -12
Conditions: none
Add all N contributions.
Rule: exact_rational_arithmetic · Uses: G001, S129
Before
affine integral primitive coefficient1 := div(8, 1)
After
affine integral primitive coefficient1 = 8
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S130
Before
affine integral primitive coefficient2 := div(21, 2)
After
affine integral primitive coefficient2 = 21/2
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S131
Before
[0, 8, 21/2]
After
H′=the supplied polynomial;integrate from -2 to 1
Conditions: none
Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference.
Rule: polynomial_primitive_identity · Uses: G001, S132
Before
affine integral H(b) ascending coefficients=[0, 8, 21/2]
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, S133
Before
affine integral H(b) Horner multiply before degree1 := mul(21/2, 1)
After
affine integral H(b) Horner multiply before degree1 = 21/2
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S134
Before
affine integral H(b) Horner add degree1 := sum(21/2, 8)
After
affine integral H(b) Horner add degree1 = 37/2
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S135
Before
affine integral H(b) Horner multiply before degree0 := mul(37/2, 1)
After
affine integral H(b) Horner multiply before degree0 = 37/2
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S136
Before
affine integral H(b) Horner add degree0 := sum(37/2, 0)
After
affine integral H(b) Horner add degree0 = 37/2
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S137
Before
affine integral H(a) ascending coefficients=[0, 8, 21/2]
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, S138
Before
affine integral H(a) Horner multiply before degree1 := mul(21/2, -2)
After
affine integral H(a) Horner multiply before degree1 = -21
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S139
Before
affine integral H(a) Horner add degree1 := sum(-21, 8)
After
affine integral H(a) Horner add degree1 = -13
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S140
Before
affine integral H(a) Horner multiply before degree0 := mul(-13, -2)
After
affine integral H(a) Horner multiply before degree0 = 26
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S141
Before
affine integral H(a) Horner add degree0 := sum(26, 0)
After
affine integral H(a) Horner add degree0 = 26
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S142
Before
affine integral oriented moment := sub(37/2, 26)
After
affine integral oriented moment = -15/2
Conditions: none
Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign.
Rule: exact_rational_arithmetic · Uses: G001, S143
Before
integral minus left sum := sub(-15/2, -12)
After
integral minus left sum = 9/2
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, S144
Before
I=-15/2;S_N=-12;N=21
After
signed finite-sum difference=9/2
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, S145
S146 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S145 · premise_and_result_audit — Recompute integral minus left sum with exact rational arithmetic, not rounded decimals.S144 · premise_and_result_audit — Recompute affine integral oriented moment with exact rational arithmetic, not rounded decimals.S143 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.S142 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S141 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals.S140 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S139 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S138 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.S137 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S136 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals.S135 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S132 · premise_and_result_audit — Recompute affine integral primitive coefficient2 with exact rational arithmetic, not rounded decimals.S131 · premise_and_result_audit — Recompute affine integral primitive coefficient1 with exact rational arithmetic, not rounded decimals.S130 · premise_and_result_audit — Recompute finite left sum with exact rational arithmetic, not rounded decimals.S129 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S128 · premise_and_result_audit — Recompute left sample20 rectangle with exact rational arithmetic, not rounded decimals.S127 · premise_and_result_audit — Recompute left sample20 value with exact rational arithmetic, not rounded decimals.S126 · premise_and_result_audit — Recompute left sample20 mx with exact rational arithmetic, not rounded decimals.S125 · premise_and_result_audit — Recompute left sample20 x with exact rational arithmetic, not rounded decimals.S124 · premise_and_result_audit — Recompute left sample20 offset with exact rational arithmetic, not rounded decimals.S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S122 · premise_and_result_audit — Recompute left sample19 rectangle with exact rational arithmetic, not rounded decimals.S121 · premise_and_result_audit — Recompute left sample19 value with exact rational arithmetic, not rounded decimals.S120 · premise_and_result_audit — Recompute left sample19 mx with exact rational arithmetic, not rounded decimals.S119 · premise_and_result_audit — Recompute left sample19 x with exact rational arithmetic, not rounded decimals.S118 · premise_and_result_audit — Recompute left sample19 offset with exact rational arithmetic, not rounded decimals.S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S116 · premise_and_result_audit — Recompute left sample18 rectangle with exact rational arithmetic, not rounded decimals.S115 · premise_and_result_audit — Recompute left sample18 value with exact rational arithmetic, not rounded decimals.S114 · premise_and_result_audit — Recompute left sample18 mx with exact rational arithmetic, not rounded decimals.S113 · premise_and_result_audit — Recompute left sample18 x with exact rational arithmetic, not rounded decimals.S112 · premise_and_result_audit — Recompute left sample18 offset with exact rational arithmetic, not rounded decimals.S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S110 · premise_and_result_audit — Recompute left sample17 rectangle with exact rational arithmetic, not rounded decimals.S109 · premise_and_result_audit — Recompute left sample17 value with exact rational arithmetic, not rounded decimals.S108 · premise_and_result_audit — Recompute left sample17 mx with exact rational arithmetic, not rounded decimals.S107 · premise_and_result_audit — Recompute left sample17 x with exact rational arithmetic, not rounded decimals.S106 · premise_and_result_audit — Recompute left sample17 offset with exact rational arithmetic, not rounded decimals.S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S104 · premise_and_result_audit — Recompute left sample16 rectangle with exact rational arithmetic, not rounded decimals.S103 · premise_and_result_audit — Recompute left sample16 value with exact rational arithmetic, not rounded decimals.S102 · premise_and_result_audit — Recompute left sample16 mx with exact rational arithmetic, not rounded decimals.S101 · premise_and_result_audit — Recompute left sample16 x with exact rational arithmetic, not rounded decimals.S100 · premise_and_result_audit — Recompute left sample16 offset with exact rational arithmetic, not rounded decimals.S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S098 · premise_and_result_audit — Recompute left sample15 rectangle with exact rational arithmetic, not rounded decimals.S097 · premise_and_result_audit — Recompute left sample15 value with exact rational arithmetic, not rounded decimals.S096 · premise_and_result_audit — Recompute left sample15 mx with exact rational arithmetic, not rounded decimals.S095 · premise_and_result_audit — Recompute left sample15 x with exact rational arithmetic, not rounded decimals.S094 · premise_and_result_audit — Recompute left sample15 offset with exact rational arithmetic, not rounded decimals.S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S092 · premise_and_result_audit — Recompute left sample14 rectangle with exact rational arithmetic, not rounded decimals.S091 · premise_and_result_audit — Recompute left sample14 value with exact rational arithmetic, not rounded decimals.S090 · premise_and_result_audit — Recompute left sample14 mx with exact rational arithmetic, not rounded decimals.S089 · premise_and_result_audit — Recompute left sample14 x with exact rational arithmetic, not rounded decimals.S088 · premise_and_result_audit — Recompute left sample14 offset with exact rational arithmetic, not rounded decimals.S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S086 · premise_and_result_audit — Recompute left sample13 rectangle with exact rational arithmetic, not rounded decimals.S085 · premise_and_result_audit — Recompute left sample13 value with exact rational arithmetic, not rounded decimals.S084 · premise_and_result_audit — Recompute left sample13 mx with exact rational arithmetic, not rounded decimals.S083 · premise_and_result_audit — Recompute left sample13 x with exact rational arithmetic, not rounded decimals.S082 · premise_and_result_audit — Recompute left sample13 offset with exact rational arithmetic, not rounded decimals.S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S080 · premise_and_result_audit — Recompute left sample12 rectangle with exact rational arithmetic, not rounded decimals.S079 · premise_and_result_audit — Recompute left sample12 value with exact rational arithmetic, not rounded decimals.S078 · premise_and_result_audit — Recompute left sample12 mx with exact rational arithmetic, not rounded decimals.S077 · premise_and_result_audit — Recompute left sample12 x with exact rational arithmetic, not rounded decimals.S076 · premise_and_result_audit — Recompute left sample12 offset with exact rational arithmetic, not rounded decimals.S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S074 · premise_and_result_audit — Recompute left sample11 rectangle with exact rational arithmetic, not rounded decimals.S073 · premise_and_result_audit — Recompute left sample11 value with exact rational arithmetic, not rounded decimals.S072 · premise_and_result_audit — Recompute left sample11 mx with exact rational arithmetic, not rounded decimals.S071 · premise_and_result_audit — Recompute left sample11 x with exact rational arithmetic, not rounded decimals.S070 · premise_and_result_audit — Recompute left sample11 offset with exact rational arithmetic, not rounded decimals.S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S068 · premise_and_result_audit — Recompute left sample10 rectangle with exact rational arithmetic, not rounded decimals.S067 · premise_and_result_audit — Recompute left sample10 value with exact rational arithmetic, not rounded decimals.S066 · premise_and_result_audit — Recompute left sample10 mx with exact rational arithmetic, not rounded decimals.S065 · premise_and_result_audit — Recompute left sample10 x with exact rational arithmetic, not rounded decimals.S064 · premise_and_result_audit — Recompute left sample10 offset with exact rational arithmetic, not rounded decimals.S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S062 · premise_and_result_audit — Recompute left sample9 rectangle with exact rational arithmetic, not rounded decimals.S061 · premise_and_result_audit — Recompute left sample9 value with exact rational arithmetic, not rounded decimals.S060 · premise_and_result_audit — Recompute left sample9 mx with exact rational arithmetic, not rounded decimals.S059 · premise_and_result_audit — Recompute left sample9 x with exact rational arithmetic, not rounded decimals.S058 · premise_and_result_audit — Recompute left sample9 offset with exact rational arithmetic, not rounded decimals.S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S056 · premise_and_result_audit — Recompute left sample8 rectangle with exact rational arithmetic, not rounded decimals.S055 · premise_and_result_audit — Recompute left sample8 value with exact rational arithmetic, not rounded decimals.S054 · premise_and_result_audit — Recompute left sample8 mx with exact rational arithmetic, not rounded decimals.S053 · premise_and_result_audit — Recompute left sample8 x with exact rational arithmetic, not rounded decimals.S052 · premise_and_result_audit — Recompute left sample8 offset 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.9/2
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: f78b389290b4c722060e58f98a03b77a0e5049e17726453210931554cf4c29d6candidate record SHA-256: b2f6d0685ee603d93658ae5d151a5f6966a914228f8059f763f03a607df8b89asemantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED