Properties of definite integrals
Problem 4.950 · easy
Suppose \( \displaystyle \int_0^{3} f(x)\,dx = -7 \), \( \displaystyle \int_0^{1} f(x)\,dx = 3 \), \( \displaystyle \int_0^{3} g(x)\,dx = 8 \) and \( \displaystyle \int_0^{1} g(x)\,dx = 6 \). Find \( \displaystyle \int_{1}^{3} \left(g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; ∫ from 1 to 3 = ∫ from 0 to 3 − ∫ from 0 to 1.
- \[ 0 \left(-10\right) + 0 \cdot 2 + 1 \cdot 2 = 2 \]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 second sentence incorrectly applies arithmetic; it uses nonsensical coefficients (0*(-10)+0*2+1*2) instead of simply computing ∫₀³g−∫₀¹g = 8−6 = 2. The setup in sentence 1 is fine, but the conclusion is derived incorrectly.qwen3.6:27b-mlx: fail (error) — The solution fails to use the given value for the integral of f(x), which is irrelevant but suggests a copy-paste error or confusion. More critically, the equation in step 2 contains arbitrary constants (-10, 2) that do not correspond to the given integrals of g(x) (8 and 6), making the derivation nonsensical even though the final answer happens to be correct.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-12 — The second sentence incorrectly applies arithmetic; it uses nonsensical coefficients (0*(-10)+0*2+1*2) instead of simply computing ∫₀³g−∫₀¹g = 8−6 = 2. The setup in sentence 1 is fine, but the conclusion is derived incorrectly.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to use the given value for the integral of f(x), which is irrelevant but suggests a copy-paste error or confusion. More critically, the equation in step 2 contains arbitrary constants (-10, 2) that do not correspond to the given integrals of g(x) (8 and 6), making the derivation nonsensical even though the final answer happens to be correct.gpt-oss:20b: fail (misleading) 2026-10-12 — The second sentence gives an unclear, incorrect algebraic expression (0*(-10)+0*2+1*2) that does not properly explain how the known integrals combine to give 2. It misleads a student about the reasoning behind the result.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to apply the linearity property of integrals to separate the integral of g(x) from the given information. It also presents an unexplained arithmetic equation that does not follow from the problem statement or standard calculus theorems.
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.