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R0006-03867 · q-20261007-advance-20261009-r06-definite-integral-0067
On all R, f(x)=(17)x+(4). On a=-2≤b=3, use N=17 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)=(17)x+(4). On a=-2≤b=3, use N=17 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, -2)
After
interval width = 5
Conditions: none
Compute the nonnegative interval width,allowing zero width.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
partition step := div(5, 17)
After
partition step = 5/17
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, 5/17)
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(17, -2)
After
left sample0 mx = -34
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
left sample0 value := sum(-34, 4)
After
left sample0 value = -30
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
left sample0 rectangle := mul(-30, 5/17)
After
left sample0 rectangle = -150/17
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
[-2, -30, -150/17]
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, 5/17)
After
left sample1 offset = 5/17
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, 5/17)
After
left sample1 x = -29/17
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
left sample1 mx := mul(17, -29/17)
After
left sample1 mx = -29
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
left sample1 value := sum(-29, 4)
After
left sample1 value = -25
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
left sample1 rectangle := mul(-25, 5/17)
After
left sample1 rectangle = -125/17
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
[-29/17, -25, -125/17]
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, 5/17)
After
left sample2 offset = 10/17
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, 10/17)
After
left sample2 x = -24/17
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
left sample2 mx := mul(17, -24/17)
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, 4)
After
left sample2 value = -20
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
left sample2 rectangle := mul(-20, 5/17)
After
left sample2 rectangle = -100/17
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
[-24/17, -20, -100/17]
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, 5/17)
After
left sample3 offset = 15/17
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, 15/17)
After
left sample3 x = -19/17
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
left sample3 mx := mul(17, -19/17)
After
left sample3 mx = -19
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
left sample3 value := sum(-19, 4)
After
left sample3 value = -15
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
left sample3 rectangle := mul(-15, 5/17)
After
left sample3 rectangle = -75/17
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
[-19/17, -15, -75/17]
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, 5/17)
After
left sample4 offset = 20/17
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, 20/17)
After
left sample4 x = -14/17
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
left sample4 mx := mul(17, -14/17)
After
left sample4 mx = -14
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
left sample4 value := sum(-14, 4)
After
left sample4 value = -10
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
left sample4 rectangle := mul(-10, 5/17)
After
left sample4 rectangle = -50/17
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
[-14/17, -10, -50/17]
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, 5/17)
After
left sample5 offset = 25/17
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, 25/17)
After
left sample5 x = -9/17
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
left sample5 mx := mul(17, -9/17)
After
left sample5 mx = -9
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
left sample5 value := sum(-9, 4)
After
left sample5 value = -5
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
left sample5 rectangle := mul(-5, 5/17)
After
left sample5 rectangle = -25/17
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
[-9/17, -5, -25/17]
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, 5/17)
After
left sample6 offset = 30/17
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, 30/17)
After
left sample6 x = -4/17
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
left sample6 mx := mul(17, -4/17)
After
left sample6 mx = -4
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
left sample6 value := sum(-4, 4)
After
left sample6 value = 0
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
left sample6 rectangle := mul(0, 5/17)
After
left sample6 rectangle = 0
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
[-4/17, 0, 0]
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, 5/17)
After
left sample7 offset = 35/17
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, 35/17)
After
left sample7 x = 1/17
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
left sample7 mx := mul(17, 1/17)
After
left sample7 mx = 1
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
left sample7 value := sum(1, 4)
After
left sample7 value = 5
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
left sample7 rectangle := mul(5, 5/17)
After
left sample7 rectangle = 25/17
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
[1/17, 5, 25/17]
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, 5/17)
After
left sample8 offset = 40/17
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, 40/17)
After
left sample8 x = 6/17
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
left sample8 mx := mul(17, 6/17)
After
left sample8 mx = 6
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
left sample8 value := sum(6, 4)
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, 5/17)
After
left sample8 rectangle = 50/17
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
[6/17, 10, 50/17]
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, 5/17)
After
left sample9 offset = 45/17
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, 45/17)
After
left sample9 x = 11/17
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
left sample9 mx := mul(17, 11/17)
After
left sample9 mx = 11
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S059
Before
left sample9 value := sum(11, 4)
After
left sample9 value = 15
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S060
Before
left sample9 rectangle := mul(15, 5/17)
After
left sample9 rectangle = 75/17
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S061
Before
[11/17, 15, 75/17]
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, 5/17)
After
left sample10 offset = 50/17
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, 50/17)
After
left sample10 x = 16/17
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S064
Before
left sample10 mx := mul(17, 16/17)
After
left sample10 mx = 16
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S065
Before
left sample10 value := sum(16, 4)
After
left sample10 value = 20
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S066
Before
left sample10 rectangle := mul(20, 5/17)
After
left sample10 rectangle = 100/17
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S067
Before
[16/17, 20, 100/17]
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, 5/17)
After
left sample11 offset = 55/17
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, 55/17)
After
left sample11 x = 21/17
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S070
Before
left sample11 mx := mul(17, 21/17)
After
left sample11 mx = 21
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S071
Before
left sample11 value := sum(21, 4)
After
left sample11 value = 25
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S072
Before
left sample11 rectangle := mul(25, 5/17)
After
left sample11 rectangle = 125/17
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S073
Before
[21/17, 25, 125/17]
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, 5/17)
After
left sample12 offset = 60/17
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, 60/17)
After
left sample12 x = 26/17
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S076
Before
left sample12 mx := mul(17, 26/17)
After
left sample12 mx = 26
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S077
Before
left sample12 value := sum(26, 4)
After
left sample12 value = 30
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S078
Before
left sample12 rectangle := mul(30, 5/17)
After
left sample12 rectangle = 150/17
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S079
Before
[26/17, 30, 150/17]
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, 5/17)
After
left sample13 offset = 65/17
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, 65/17)
After
left sample13 x = 31/17
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S082
Before
left sample13 mx := mul(17, 31/17)
After
left sample13 mx = 31
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S083
Before
left sample13 value := sum(31, 4)
After
left sample13 value = 35
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S084
Before
left sample13 rectangle := mul(35, 5/17)
After
left sample13 rectangle = 175/17
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S085
Before
[31/17, 35, 175/17]
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, 5/17)
After
left sample14 offset = 70/17
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, 70/17)
After
left sample14 x = 36/17
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S088
Before
left sample14 mx := mul(17, 36/17)
After
left sample14 mx = 36
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S089
Before
left sample14 value := sum(36, 4)
After
left sample14 value = 40
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S090
Before
left sample14 rectangle := mul(40, 5/17)
After
left sample14 rectangle = 200/17
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S091
Before
[36/17, 40, 200/17]
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, 5/17)
After
left sample15 offset = 75/17
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, 75/17)
After
left sample15 x = 41/17
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S094
Before
left sample15 mx := mul(17, 41/17)
After
left sample15 mx = 41
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S095
Before
left sample15 value := sum(41, 4)
After
left sample15 value = 45
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S096
Before
left sample15 rectangle := mul(45, 5/17)
After
left sample15 rectangle = 225/17
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S097
Before
[41/17, 45, 225/17]
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, 5/17)
After
left sample16 offset = 80/17
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, 80/17)
After
left sample16 x = 46/17
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S100
Before
left sample16 mx := mul(17, 46/17)
After
left sample16 mx = 46
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S101
Before
left sample16 value := sum(46, 4)
After
left sample16 value = 50
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S102
Before
left sample16 rectangle := mul(50, 5/17)
After
left sample16 rectangle = 250/17
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S103
Before
[46/17, 50, 250/17]
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
finite left sum := sum(-150/17, -125/17, -100/17, -75/17, -50/17, -25/17, 0, 25/17, 50/17, 75/17, 100/17, 125/17, 150/17, 175/17, 200/17, 225/17, 250/17)
After
finite left sum = 50
Conditions: none
Add all N contributions.
Rule: exact_rational_arithmetic · Uses: G001, S105
Before
affine integral primitive coefficient1 := div(4, 1)
After
affine integral primitive coefficient1 = 4
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S106
Before
affine integral primitive coefficient2 := div(17, 2)
After
affine integral primitive coefficient2 = 17/2
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S107
Before
[0, 4, 17/2]
After
H′=the supplied polynomial;integrate from -2 to 3
Conditions: none
Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference.
Rule: polynomial_primitive_identity · Uses: G001, S108
Before
affine integral H(b) ascending coefficients=[0, 4, 17/2]
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, S109
Before
affine integral H(b) Horner multiply before degree1 := mul(17/2, 3)
After
affine integral H(b) Horner multiply before degree1 = 51/2
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S110
Before
affine integral H(b) Horner add degree1 := sum(51/2, 4)
After
affine integral H(b) Horner add degree1 = 59/2
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S111
Before
affine integral H(b) Horner multiply before degree0 := mul(59/2, 3)
After
affine integral H(b) Horner multiply before degree0 = 177/2
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S112
Before
affine integral H(b) Horner add degree0 := sum(177/2, 0)
After
affine integral H(b) Horner add degree0 = 177/2
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S113
Before
affine integral H(a) ascending coefficients=[0, 4, 17/2]
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, S114
Before
affine integral H(a) Horner multiply before degree1 := mul(17/2, -2)
After
affine integral H(a) Horner multiply before degree1 = -17
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S115
Before
affine integral H(a) Horner add degree1 := sum(-17, 4)
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, S116
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, S117
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, S118
Before
affine integral oriented moment := sub(177/2, 26)
After
affine integral oriented moment = 125/2
Conditions: none
Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign.
Rule: exact_rational_arithmetic · Uses: G001, S119
Before
integral minus left sum := sub(125/2, 50)
After
integral minus left sum = 25/2
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, S120
Before
I=125/2;S_N=50;N=17
After
signed finite-sum difference=25/2
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, S121
S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S121 · premise_and_result_audit — Recompute integral minus left sum with exact rational arithmetic, not rounded decimals.S120 · premise_and_result_audit — Recompute affine integral oriented moment with exact rational arithmetic, not rounded decimals.S119 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.S118 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S117 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals.S116 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S114 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.S113 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S112 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals.S111 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S109 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S108 · premise_and_result_audit — Recompute affine integral primitive coefficient2 with exact rational arithmetic, not rounded decimals.S107 · premise_and_result_audit — Recompute affine integral primitive coefficient1 with exact rational arithmetic, not rounded decimals.S106 · premise_and_result_audit — Recompute finite left sum 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.25/2
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: d3f5777b92545634b0f562ddb755ab18dcd8691ec4599120f7d2f7fac776cf20candidate record SHA-256: ccfd002509e69fb27c30fefb23c2ceb7f44d09fb9fb8a03a3e9be753ec62fa37semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED