Properties of definite integrals
Problem 4.329 · easy
Suppose \( \displaystyle \int_0^{6} f(x)\,dx = 4 \), \( \displaystyle \int_0^{3} f(x)\,dx = 5 \), \( \displaystyle \int_0^{6} g(x)\,dx = 2 \) and \( \displaystyle \int_0^{3} g(x)\,dx = -4 \). Find \( \displaystyle \int_{6}^{3} \left(f(x) + g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; ∫ from 3 to 6 = ∫ from 0 to 6 − ∫ from 0 to 3; swapping the limits changes the sign.
- \[ 1 \left(-6\right) + 0 \left(-3\right) + 1 \cdot 1 = -5 \]Combine the known values.✓ Proved
Answer \( -5 \)
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) — The second sentence gives a nonsensical equation 1*(-6)+0*(-3)+1*1=-5 that does not follow from the given data. It does not correctly combine the known integrals to obtain the result.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly calculate the intermediate integrals \int_3^6 f(x) dx and \int_3^6 g(x) dx, jumping straight to a numerical combination that is not derived from the given data in the text. While the final answer is correct, the reasoning is opaque and does not demonstrate the necessary steps.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly calculate the intermediate integrals \int_3^6 f(x) dx and \int_3^6 g(x) dx, jumping straight to a numerical combination that is not derived from the given data in the text. While the final answer is correct, the reasoning is opaque and does not demonstrate the necessary steps.gpt-oss:20b: fail (error) 2026-10-04 — The second sentence gives a nonsensical equation 1*(-6)+0*(-3)+1*1=-5 that does not follow from the given data. It does not correctly combine the known integrals to obtain the result.qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer returned a non-objectgpt-oss:20b: fail (misleading) 2026-10-04 — Sentence 2 presents an ill‑formed algebraic expression that does not correctly combine the given integrals; it obscures the correct calculation and could mislead a student about how to use the data.
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 2026-10-04 with SymPy 1.14.0.