Properties of definite integrals
Problem 4.327 · easy
Suppose \( \displaystyle \int_0^{4} f(x)\,dx = 1 \), \( \displaystyle \int_0^{2} f(x)\,dx = 6 \), \( \displaystyle \int_0^{4} g(x)\,dx = -1 \) and \( \displaystyle \int_0^{2} g(x)\,dx = -1 \). Find \( \displaystyle \int_{2}^{4} \left(g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; ∫ from 2 to 4 = ∫ from 0 to 4 − ∫ from 0 to 2.
- \[ 0 \left(-5\right) + 1 \cdot 0 + 0 \cdot 2 = 0 \]Combine the known values.✓ Proved
Answer \( 0 \)
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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution’s second line contains a nonsensical algebraic expression that does not follow from the given data. It incorrectly mixes unrelated numbers anqwen3.6:27b-mlx: fail (error) — The solution contains nonsensical arithmetic (0*(-5) + 1*0 + 0*2) that does not correspond to the given integral values. The correct calculation is \int_2^4 g(x) dx = \int_0^4 g(x) dx - \int_0^2 g(x) dx = -1 - (-1) = 0, but the provided steps are mathematically incoherent.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution contains nonsensical arithmetic (0*(-5) + 1*0 + 0*2) that does not correspond to the given integral values. The correct calculation is \int_2^4 g(x) dx = \int_0^4 g(x) dx - \int_0^2 g(x) dx = -1 - (-1) = 0, but the provided steps are mathematically incoherent.gpt-oss:20b: inconclusive 2026-10-04 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution’s second line contains a nonsensical algebraic expression that does not follow from the given data. It incorrectly mixes unrelated numbers anqwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to actually compute the integral of g(x) using the given values. It provides a nonsensical arithmetic expression '0*(-5) + 1*0 + 0*2' that does not correspond to the problem data (where int_0^4 g = -1 and int_0^2 g = -1, so the answer should be 0, but the derivation is completely missing and the numbers used are wrong).gpt-oss:20b: fail (misleading) 2026-10-04 — The second sentence presents an incorrect algebraic expression that does not reflect the proper computation of ∫₂⁴ g(x)dx = ∫₀⁴ g(x)dx – ∫₀² g(x)dx = (‑1) – (‑1) = 0. The reasoning is therefore misleading.
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.