Properties of definite integrals
Problem 4.492 · easy
Suppose \( \displaystyle \int_0^{3} f(x)\,dx = -5 \), \( \displaystyle \int_0^{2} f(x)\,dx = -5 \), \( \displaystyle \int_0^{3} g(x)\,dx = 9 \) and \( \displaystyle \int_0^{2} g(x)\,dx = 7 \). Find \( \displaystyle \int_{3}^{0} \left(2 f(x) - 3 g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; swapping the limits changes the sign.
- \[ 0 \left(-3\right) + 2 \cdot 5 - 3 \left(-9\right) = 37 \]Combine the known values.✓ Proved
Answer \( 37 \)
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 line incorrectly includes a term 0*(-3) and does not correctly justify the use of the given integrals. The correct computation should use ∫₃⁰f = 5 and ∫₃⁰g = -9, leading to 2·5 – 3·(-9) = 37. The extraneous 0*(-3) term makes the reasoning invalid.qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-06 — The second line incorrectly includes a term 0*(-3) and does not correctly justify the use of the given integrals. The correct computation should use ∫₃⁰f = 5 and ∫₃⁰g = -9, leading to 2·5 – 3·(-9) = 37. The extraneous 0*(-3) term makes the reasoning invalid.qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer returned a non-objectgpt-oss:20b: fail (misleading) 2026-10-06 — The algebraic step 0*(-3)+2*5-3*(-9)=37 is incorrect: it misuses the given integrals (signs and values). While the final numeric result happens to be right, the reasoning is flawed and would mislead a student.qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer returned a non-object
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 2026-10-06 with SymPy 1.14.0.