Explore related knowledge and questions →
← Question derivation index · Original question batch
R0006-03870 · q-20261007-advance-20261009-r06-definite-integral-0070
On all R, f(x)=(-20)x+(7). On a=1≤b=3, use N=20 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)=(-20)x+(7). On a=1≤b=3, use N=20 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, 20)
After
partition step = 1/10
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, 1/10)
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(-20, 1)
After
left sample0 mx = -20
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
left sample0 value := sum(-20, 7)
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, 1/10)
After
left sample0 rectangle = -13/10
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, -13/10]
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, 1/10)
After
left sample1 offset = 1/10
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, 1/10)
After
left sample1 x = 11/10
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
left sample1 mx := mul(-20, 11/10)
After
left sample1 mx = -22
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
left sample1 value := sum(-22, 7)
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, 1/10)
After
left sample1 rectangle = -3/2
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
[11/10, -15, -3/2]
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, 1/10)
After
left sample2 offset = 1/5
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, 1/5)
After
left sample2 x = 6/5
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
left sample2 mx := mul(-20, 6/5)
After
left sample2 mx = -24
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
left sample2 value := sum(-24, 7)
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, 1/10)
After
left sample2 rectangle = -17/10
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
[6/5, -17, -17/10]
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, 1/10)
After
left sample3 offset = 3/10
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, 3/10)
After
left sample3 x = 13/10
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
left sample3 mx := mul(-20, 13/10)
After
left sample3 mx = -26
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
left sample3 value := sum(-26, 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, 1/10)
After
left sample3 rectangle = -19/10
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
[13/10, -19, -19/10]
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, 1/10)
After
left sample4 offset = 2/5
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, 2/5)
After
left sample4 x = 7/5
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
left sample4 mx := mul(-20, 7/5)
After
left sample4 mx = -28
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
left sample4 value := sum(-28, 7)
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, 1/10)
After
left sample4 rectangle = -21/10
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
[7/5, -21, -21/10]
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, 1/10)
After
left sample5 offset = 1/2
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, 1/2)
After
left sample5 x = 3/2
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
left sample5 mx := mul(-20, 3/2)
After
left sample5 mx = -30
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
left sample5 value := sum(-30, 7)
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, 1/10)
After
left sample5 rectangle = -23/10
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
[3/2, -23, -23/10]
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, 1/10)
After
left sample6 offset = 3/5
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, 3/5)
After
left sample6 x = 8/5
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
left sample6 mx := mul(-20, 8/5)
After
left sample6 mx = -32
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
left sample6 value := sum(-32, 7)
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, 1/10)
After
left sample6 rectangle = -5/2
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
[8/5, -25, -5/2]
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, 1/10)
After
left sample7 offset = 7/10
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, 7/10)
After
left sample7 x = 17/10
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
left sample7 mx := mul(-20, 17/10)
After
left sample7 mx = -34
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
left sample7 value := sum(-34, 7)
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, 1/10)
After
left sample7 rectangle = -27/10
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
[17/10, -27, -27/10]
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, 1/10)
After
left sample8 offset = 4/5
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(1, 4/5)
After
left sample8 x = 9/5
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
left sample8 mx := mul(-20, 9/5)
After
left sample8 mx = -36
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
left sample8 value := sum(-36, 7)
After
left sample8 value = -29
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S054
Before
left sample8 rectangle := mul(-29, 1/10)
After
left sample8 rectangle = -29/10
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
[9/5, -29, -29/10]
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, 1/10)
After
left sample9 offset = 9/10
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(1, 9/10)
After
left sample9 x = 19/10
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
left sample9 mx := mul(-20, 19/10)
After
left sample9 mx = -38
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S059
Before
left sample9 value := sum(-38, 7)
After
left sample9 value = -31
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S060
Before
left sample9 rectangle := mul(-31, 1/10)
After
left sample9 rectangle = -31/10
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S061
Before
[19/10, -31, -31/10]
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, 1/10)
After
left sample10 offset = 1
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(1, 1)
After
left sample10 x = 2
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S064
Before
left sample10 mx := mul(-20, 2)
After
left sample10 mx = -40
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S065
Before
left sample10 value := sum(-40, 7)
After
left sample10 value = -33
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S066
Before
left sample10 rectangle := mul(-33, 1/10)
After
left sample10 rectangle = -33/10
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S067
Before
[2, -33, -33/10]
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, 1/10)
After
left sample11 offset = 11/10
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(1, 11/10)
After
left sample11 x = 21/10
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S070
Before
left sample11 mx := mul(-20, 21/10)
After
left sample11 mx = -42
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S071
Before
left sample11 value := sum(-42, 7)
After
left sample11 value = -35
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S072
Before
left sample11 rectangle := mul(-35, 1/10)
After
left sample11 rectangle = -7/2
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S073
Before
[21/10, -35, -7/2]
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, 1/10)
After
left sample12 offset = 6/5
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(1, 6/5)
After
left sample12 x = 11/5
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S076
Before
left sample12 mx := mul(-20, 11/5)
After
left sample12 mx = -44
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S077
Before
left sample12 value := sum(-44, 7)
After
left sample12 value = -37
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S078
Before
left sample12 rectangle := mul(-37, 1/10)
After
left sample12 rectangle = -37/10
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S079
Before
[11/5, -37, -37/10]
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, 1/10)
After
left sample13 offset = 13/10
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(1, 13/10)
After
left sample13 x = 23/10
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S082
Before
left sample13 mx := mul(-20, 23/10)
After
left sample13 mx = -46
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S083
Before
left sample13 value := sum(-46, 7)
After
left sample13 value = -39
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S084
Before
left sample13 rectangle := mul(-39, 1/10)
After
left sample13 rectangle = -39/10
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S085
Before
[23/10, -39, -39/10]
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, 1/10)
After
left sample14 offset = 7/5
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(1, 7/5)
After
left sample14 x = 12/5
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S088
Before
left sample14 mx := mul(-20, 12/5)
After
left sample14 mx = -48
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S089
Before
left sample14 value := sum(-48, 7)
After
left sample14 value = -41
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S090
Before
left sample14 rectangle := mul(-41, 1/10)
After
left sample14 rectangle = -41/10
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S091
Before
[12/5, -41, -41/10]
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, 1/10)
After
left sample15 offset = 3/2
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(1, 3/2)
After
left sample15 x = 5/2
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S094
Before
left sample15 mx := mul(-20, 5/2)
After
left sample15 mx = -50
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S095
Before
left sample15 value := sum(-50, 7)
After
left sample15 value = -43
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S096
Before
left sample15 rectangle := mul(-43, 1/10)
After
left sample15 rectangle = -43/10
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S097
Before
[5/2, -43, -43/10]
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, 1/10)
After
left sample16 offset = 8/5
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(1, 8/5)
After
left sample16 x = 13/5
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S100
Before
left sample16 mx := mul(-20, 13/5)
After
left sample16 mx = -52
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S101
Before
left sample16 value := sum(-52, 7)
After
left sample16 value = -45
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S102
Before
left sample16 rectangle := mul(-45, 1/10)
After
left sample16 rectangle = -9/2
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S103
Before
[13/5, -45, -9/2]
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, 1/10)
After
left sample17 offset = 17/10
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(1, 17/10)
After
left sample17 x = 27/10
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S106
Before
left sample17 mx := mul(-20, 27/10)
After
left sample17 mx = -54
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S107
Before
left sample17 value := sum(-54, 7)
After
left sample17 value = -47
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S108
Before
left sample17 rectangle := mul(-47, 1/10)
After
left sample17 rectangle = -47/10
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S109
Before
[27/10, -47, -47/10]
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, 1/10)
After
left sample18 offset = 9/5
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(1, 9/5)
After
left sample18 x = 14/5
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S112
Before
left sample18 mx := mul(-20, 14/5)
After
left sample18 mx = -56
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S113
Before
left sample18 value := sum(-56, 7)
After
left sample18 value = -49
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S114
Before
left sample18 rectangle := mul(-49, 1/10)
After
left sample18 rectangle = -49/10
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S115
Before
[14/5, -49, -49/10]
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, 1/10)
After
left sample19 offset = 19/10
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(1, 19/10)
After
left sample19 x = 29/10
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S118
Before
left sample19 mx := mul(-20, 29/10)
After
left sample19 mx = -58
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S119
Before
left sample19 value := sum(-58, 7)
After
left sample19 value = -51
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S120
Before
left sample19 rectangle := mul(-51, 1/10)
After
left sample19 rectangle = -51/10
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S121
Before
[29/10, -51, -51/10]
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
finite left sum := sum(-13/10, -3/2, -17/10, -19/10, -21/10, -23/10, -5/2, -27/10, -29/10, -31/10, -33/10, -7/2, -37/10, -39/10, -41/10, -43/10, -9/2, -47/10, -49/10, -51/10)
After
finite left sum = -64
Conditions: none
Add all N contributions.
Rule: exact_rational_arithmetic · Uses: G001, S123
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, S124
Before
affine integral primitive coefficient2 := div(-20, 2)
After
affine integral primitive coefficient2 = -10
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S125
Before
[0, 7, -10]
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, S126
Before
affine integral H(b) ascending coefficients=[0, 7, -10]
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, S127
Before
affine integral H(b) Horner multiply before degree1 := mul(-10, 3)
After
affine integral H(b) Horner multiply before degree1 = -30
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S128
Before
affine integral H(b) Horner add degree1 := sum(-30, 7)
After
affine integral H(b) Horner add degree1 = -23
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S129
Before
affine integral H(b) Horner multiply before degree0 := mul(-23, 3)
After
affine integral H(b) Horner multiply before degree0 = -69
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S130
Before
affine integral H(b) Horner add degree0 := sum(-69, 0)
After
affine integral H(b) Horner add degree0 = -69
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S131
Before
affine integral H(a) ascending coefficients=[0, 7, -10]
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, S132
Before
affine integral H(a) Horner multiply before degree1 := mul(-10, 1)
After
affine integral H(a) Horner multiply before degree1 = -10
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S133
Before
affine integral H(a) Horner add degree1 := sum(-10, 7)
After
affine integral H(a) Horner add degree1 = -3
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S134
Before
affine integral H(a) Horner multiply before degree0 := mul(-3, 1)
After
affine integral H(a) Horner multiply before degree0 = -3
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S135
Before
affine integral H(a) Horner add degree0 := sum(-3, 0)
After
affine integral H(a) Horner add degree0 = -3
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S136
Before
affine integral oriented moment := sub(-69, -3)
After
affine integral oriented moment = -66
Conditions: none
Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign.
Rule: exact_rational_arithmetic · Uses: G001, S137
Before
integral minus left sum := sub(-66, -64)
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, S138
Before
I=-66;S_N=-64;N=20
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, S139
S140 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S139 · premise_and_result_audit — Recompute integral minus left sum with exact rational arithmetic, not rounded decimals.S138 · premise_and_result_audit — Recompute affine integral oriented moment with exact rational arithmetic, not rounded decimals.S137 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.S136 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S135 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals.S134 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.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 H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.S131 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S130 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals.S129 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S126 · premise_and_result_audit — Recompute affine integral primitive coefficient2 with exact rational arithmetic, not rounded decimals.S125 · premise_and_result_audit — Recompute affine integral primitive coefficient1 with exact rational arithmetic, not rounded decimals.S124 · premise_and_result_audit — Recompute finite left sum 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.-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: d63214579246b52a4e5b8b38a27bd2a91fceec957f10d1989f1c8bb8beea56b1candidate record SHA-256: 51974df6f797aa18c4050491ce6eacd55786e04e6e5a517c26999db7bee17fe1semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED