Properties of definite integrals
Problem 4.945 · easy
Suppose \( \displaystyle \int_0^{6} f(x)\,dx = -1 \), \( \displaystyle \int_0^{1} f(x)\,dx = -7 \), \( \displaystyle \int_0^{6} g(x)\,dx = 7 \) and \( \displaystyle \int_0^{1} g(x)\,dx = -3 \). Find \( \displaystyle \int_{6}^{1} \left(f(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; ∫ from 1 to 6 = ∫ from 0 to 6 − ∫ from 0 to 1; swapping the limits changes the sign.
- \[ 1 \left(-6\right) + 0 \left(-10\right) + 0 \left(-5\right) = -6 \]Combine the known values.✓ Proved
Answer \( -6 \)
Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | integrated concrete linear functions built to have the given integrals |
Reviewers
gpt-oss:20b: fail (error) — Sentence 2 contains an unexplained linear combination (1*(-6)+0*(-10)+0*(-5)) that does not follow from any stated facts. It is unclear what the coefficients represent and how the known integrals are combined. This step is incorrect and misleading.qwen3.6:27b-mlx: fail (error) — The solution calculates the integral of f from 1 to 6, but the problem asks for the integral from 6 to 1. The final answer should be 6, not -6.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-12 — Sentence 2 contains an unexplained linear combination (1*(-6)+0*(-10)+0*(-5)) that does not follow from any stated facts. It is unclear what the coefficients represent and how the known integrals are combined. This step is incorrect and misleading.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution calculates the integral of f from 1 to 6, but the problem asks for the integral from 6 to 1. The final answer should be 6, not -6.gpt-oss:20b: fail (misleading) 2026-10-12 — The second sentence presents an arbitrary linear combination of numbers (1,0,0) with values (-6,-10,-5) that is not justified by the problem data. It gives a misleading impression that the answer comes from a weighted sum of unrelated integrals, whereas the correct reasoning is simply ∫_6^1 f = -∫_1^6 f = -(∫_0^6 f - ∫_0^1 f) = -(-1 - (-7)) = -6.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to address the actual question, which asks for the integral of f(x) only, not a combination of f and g. Furthermore, the arithmetic in the equation line is nonsensical and does not follow from the stated properties.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/integral_properties, checked 2026-10-12 with SymPy 1.14.0.