Properties of definite integrals
Problem 4.626 · easy
Suppose \( \displaystyle \int_0^{6} f(x)\,dx = 3 \), \( \displaystyle \int_0^{3} f(x)\,dx = -8 \), \( \displaystyle \int_0^{6} g(x)\,dx = 7 \) and \( \displaystyle \int_0^{3} g(x)\,dx = 7 \). Find \( \displaystyle \int_{6}^{3} \left(2 f(x) - 3 g(x)\right) dx \).
- 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.
- \[ 2 \left(-11\right) - 3 \cdot 0 + 0 \left(-3\right) = -22 \]Combine the known values.✓ Proved
Answer \( -22 \)
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 incorrectly computes \(\int_{3}^{6}f(x)\,dx\). Since \(\int_{0}^{6}f=3\) and \(\int_{0}^{3}f=-8\), we have \(\int_{3}^{6}f=3-(-8)=11\), not \qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
gpt-oss:20b: inconclusive 2026-10-07 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution incorrectly computes \(\int_{3}^{6}f(x)\,dx\). Since \(\int_{0}^{6}f=3\) and \(\int_{0}^{3}f=-8\), we have \(\int_{3}^{6}f=3-(-8)=11\), not \qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer returned a non-objectqwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution contains nonsensical arithmetic in the final step (e.g., '0*(-3)'), and the intermediate values used (-11 and 0) are not derived or explained, making the logic opaque and incorrect.gpt-oss:20b: fail (error) 2026-10-07 — The solution incorrectly uses ∫₃⁶f(x) = –11 instead of 11, leading to an incorrect sign in the calculation. The final result should be –22, but the reasoning is flawed.
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 2026-10-07 with SymPy 1.14.0.