# Question 3850: Integer Endpoint Recovered from a Nonzero Constant Integral

[Return to catalogue page 20](https://ocme.evemisslab.com/en/questions/page/20/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261009-r06-definite-integral-foundations/#q-20261007-advance-20261009-r06-definite-integral-0100)

This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0006-03850 · q-20261007-advance-20261009-r06-definite-integral-0100



## Original question and givens

On all R, the constant integrand is c=25≠0. Starting at a=12, its oriented integral has target -625/1. Recover b, which is promised to remain an integer endpoint in this slice. Use reduced positive-denominator fractions, retaining /1 for integers.


- G001 — On all R, the constant integrand is c=25≠0. Starting at a=12, its oriented integral has target -625/1. Recover b, which is promised to remain an integer endpoint in this slice. Use reduced positive-denominator fractions, retaining /1 for integers.





## Definition and scope

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.


- [mko-definite-integral](https://ocme.evemisslab.com/en/mko/mko-definite-integral/)





## Step-by-step derivation

S001 · impliesBefore

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.

Step identifiers and dependenciesRule: polynomial_moment_scope · Uses: G001





S002 · calculationBefore

oriented interval displacement := div(-625, 25)

After

oriented interval displacement = -25

Conditions: none

Divide c(b−a)=target by nonzero c,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

recovered endpoint := sum(12, -25)

After

recovered endpoint = -13

Conditions: none

Add the oriented displacement to a;this frozen exercise slice requires an integer endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

endpoint reverse substitution := mul(25, -25)

After

endpoint reverse substitution = -625

Conditions: none

Substitute into c(b−a) to audit the target;target0 gives b=a.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · impliesBefore

c=25≠0;b=-13

After

I(a,b)=-625=target

Conditions: none

A nonzero constant makes the endpoint unique;negative c or target does not permit arbitrary endpoint reordering.

Step identifiers and dependenciesRule: unique_constant_endpoint · Uses: G001, S004









## Reverse checks


- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Recompute endpoint reverse substitution with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute recovered endpoint with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute oriented interval displacement 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.





## Result

-13/1

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 70487747c78c996e72927866d1646c4fc257af7848d43c9c157fec3ea88d98fd
- candidate record SHA-256: ac77e72c7cf8e8fcfc7fae4195678c441ac0f9bf82b5ef53cf1a52d3fac0236b
- semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d
- question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261009-r06-definite-integral-0100.json)