Properties of definite integrals
Problem 4.948 · easy
Suppose \( \displaystyle \int_0^{6} f(x)\,dx = 5 \), \( \displaystyle \int_0^{3} f(x)\,dx = 6 \), \( \displaystyle \int_0^{6} g(x)\,dx = 7 \) and \( \displaystyle \int_0^{3} g(x)\,dx = 2 \). Find \( \displaystyle \int_{0}^{3} \left(g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out.
- \[ 0 \cdot 6 + 0 \cdot 3 + 1 \cdot 2 = 2 \]Combine the known values.✓ Proved
Answer \( 2 \)
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 solution does not correctly use the given integral values; the second line is a meaningless algebraic manipulation that does not justify the result.qwen3.6:27b-mlx: fail (error) — The solution provides no logical connection between the given integrals and the final answer, instead presenting an arbitrary arithmetic calculation. It fails to simply state that the answer is given directly in the problem statement.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-12 — The solution does not correctly use the given integral values; the second line is a meaningless algebraic manipulation that does not justify the result.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution provides no logical connection between the given integrals and the final answer, instead presenting an arbitrary arithmetic calculation. It fails to simply state that the answer is given directly in the problem statement.gpt-oss:20b: fail (misleading) 2026-10-12 — The solution’s second line is nonsensical: it multiplies unrelated constants by the given integrals and claims to combine them, yet it does not use the provided data correctly. This would mislead a student into thinking the answer follows from a valid algebraic manipulation, when in fact the answer is simply given in the problem statement.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to address the actual problem, which simply asks to read off the value of a given integral. The provided equation is nonsensical and does not follow from the premises.
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.