Properties of definite integrals
Problem 4.944 · easy
Suppose \( \displaystyle \int_0^{5} f(x)\,dx = -7 \), \( \displaystyle \int_0^{4} f(x)\,dx = 6 \), \( \displaystyle \int_0^{5} g(x)\,dx = -1 \) and \( \displaystyle \int_0^{4} g(x)\,dx = -4 \). Find \( \displaystyle \int_{4}^{0} \left(f(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; swapping the limits changes the sign.
- \[ 1 \left(-6\right) + 0 \cdot 4 + 0 \left(-4\right) = -6 \]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 (error) — The solution does not correctly set up the integral ∫₄⁰f(x)dx = -∫₀⁴f(x)dx and incorrectly uses an arbitrary linear combination of the given integrals. The steps do not follow from the problem data and give no justification for the result.qwen3.6:27b-mlx: fail (error) — The solution fails to use the given information to calculate the integral. It provides a nonsensical arithmetic expression that does not derive from the problem statement or standard calculus properties.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-12 — The solution does not correctly set up the integral ∫₄⁰f(x)dx = -∫₀⁴f(x)dx and incorrectly uses an arbitrary linear combination of the given integrals. The steps do not follow from the problem data and give no justification for the result.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to use the given information to calculate the integral. It provides a nonsensical arithmetic expression that does not derive from the problem statement or standard calculus properties.gpt-oss:20b: fail (error) 2026-10-12 — The solution incorrectly uses a value of –6 as if it were known from the data. It should compute ∫₄⁰f(x) as –∫₀⁴f(x)=–6, not invoke an arbitrary –6 in an equation.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to use the additivity property of integrals to isolate the integral from 4 to 5, and instead presents an unexplained arithmetic expression that does not follow from the given data.
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.