Properties of definite integrals
Problem 4.324 · easy
Suppose \( \displaystyle \int_0^{5} f(x)\,dx = -1 \), \( \displaystyle \int_0^{2} f(x)\,dx = -6 \), \( \displaystyle \int_0^{5} g(x)\,dx = 4 \) and \( \displaystyle \int_0^{2} g(x)\,dx = 1 \). Find \( \displaystyle \int_{5}^{0} \left(f(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; swapping the limits changes the sign.
- \[ 0 \left(-4\right) + 0 \left(-5\right) + 1 \cdot 1 = 1 \]Combine the known values.✓ Proved
Answer \( 1 \)
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’s second line is nonsensical: it multiplies zeros by unrelated numbers and does not use the given integrals. The correct reasoning is simply that ∫₅⁰f(x)dx=−∫₀⁵f(x)dx=1.qwen3.6:27b-mlx: fail (error) — The solution ignores the given information about f(x) and g(x) entirely, instead performing an unexplained arithmetic calculation that does not correspond to the problem statement. The correct answer should be derived from \int_{5}^{0} f(x) dx = -\int_{0}^{5} f(x) dx = -(-1) = 1, but the provided reasoning is nonsensical.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution ignores the given information about f(x) and g(x) entirely, instead performing an unexplained arithmetic calculation that does not correspond to the problem statement. The correct answer should be derived from \int_{5}^{0} f(x) dx = -\int_{0}^{5} f(x) dx = -(-1) = 1, but the provided reasoning is nonsensical.gpt-oss:20b: fail (error) 2026-10-04 — The solution’s second line is nonsensical: it multiplies zeros by unrelated numbers and does not use the given integrals. The correct reasoning is simply that ∫₅⁰f(x)dx=−∫₀⁵f(x)dx=1.qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution completely ignores the given functions f and g and the specific integral requested. It performs an unrelated arithmetic calculation that does not follow from the problem statement.gpt-oss:20b: fail (error) 2026-10-04 — The solution’s second line is nonsensical: it multiplies zeros by arbitrary numbers and adds 1*1, which does not use the given integrals. The reasoning is incorrect even though the final numeric answer happens to be right.
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.