Properties of definite integrals
Problem 4.790 · easy
Suppose \( \displaystyle \int_0^{3} f(x)\,dx = -9 \), \( \displaystyle \int_0^{2} f(x)\,dx = 5 \), \( \displaystyle \int_0^{3} g(x)\,dx = 2 \) and \( \displaystyle \int_0^{2} g(x)\,dx = 6 \). Find \( \displaystyle \int_{2}^{0} \left(g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; swapping the limits changes the sign.
- \[ 1 \left(-6\right) + 0 \left(-5\right) + 0 \left(-2\right) = -6 \]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 (misleading) — The solution mixes unrelated integrals and constants, incorrectly applying linearity to unrelated terms. It suggests a complex combination of f and g values that is unnecessary and misleading for the simple fact that ∫₂⁰g(x)dx = -∫₀²g(x)dx = -6.qwen3.6:27b-mlx: fail (error) — The solution fails to use the fundamental property of integrals (additivity over intervals) to derive the value of the integral from 2 to 3, instead presenting an unexplained linear combination of unrelated values. It also ignores the function f(x) entirely, which is irrelevant, but the core error is the lack of logical derivation for the specific integral requested.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-09 — The solution mixes unrelated integrals and constants, incorrectly applying linearity to unrelated terms. It suggests a complex combination of f and g values that is unnecessary and misleading for the simple fact that ∫₂⁰g(x)dx = -∫₀²g(x)dx = -6.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to use the fundamental property of integrals (additivity over intervals) to derive the value of the integral from 2 to 3, instead presenting an unexplained linear combination of unrelated values. It also ignores the function f(x) entirely, which is irrelevant, but the core error is the lack of logical derivation for the specific integral requested.gpt-oss:20b: fail (error) 2026-10-09 — The solution’s equation line is nonsensical: it mixes unrelated integrals and constants. The correct reasoning is simply ∫₂⁰ g(x)dx = –∫₀² g(x)dx = –6, using only the given value ∫₀² g(x)dx = 6. The sentence about linearity and sign change is fine, but the algebraic step is incorrect.qwen3.6:27b-mlx: inconclusive 2026-10-09 — 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 2026-10-09 with SymPy 1.14.0.