Properties of definite integrals
Problem 4.493 · easy
Suppose \( \displaystyle \int_0^{6} f(x)\,dx = 3 \), \( \displaystyle \int_0^{1} f(x)\,dx = -1 \), \( \displaystyle \int_0^{6} g(x)\,dx = 1 \) and \( \displaystyle \int_0^{1} g(x)\,dx = -9 \). Find \( \displaystyle \int_{1}^{0} \left(g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out; swapping the limits changes the sign.
- \[ 0 \cdot 1 + 0 \left(-1\right) + 1 \cdot 9 = 9 \]Combine the known values.✓ Proved
Answer \( 9 \)
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 equation 0*1 + 0*(-1) + 1*9 = 9 is nonsensical and does not follow from the given data. The correct reasoning is that ∫₁⁰g(x)dx = –∫₀¹g(x)dx = –(–9) = 9. The provided steps do not correctly model the problem or apply the given integrals.qwen3.6:27b-mlx: fail (error) — The solution fails to use the given information about f(x) and g(x) to derive the answer. It presents an arbitrary arithmetic calculation (0*1 + 0*(-1) + 1*9) that does not follow from the problem statement or standard calculus theorems. The correct approach requires using the additive property of integrals: \int_0^6 g(x)dx = \int_0^1 g(x)dx + \int_1^6 g(x)dx, but the question asks for \int_1^0 g(x)dx, which is simply -\int_0^1 g(x)dx = -(-9) = 9. While the final number is correct, the reasoning provided in the solution is completely disconnected from the problem setup and mathematically unjustified.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-06 — The solution’s equation 0*1 + 0*(-1) + 1*9 = 9 is nonsensical and does not follow from the given data. The correct reasoning is that ∫₁⁰g(x)dx = –∫₀¹g(x)dx = –(–9) = 9. The provided steps do not correctly model the problem or apply the given integrals.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to use the given information about f(x) and g(x) to derive the answer. It presents an arbitrary arithmetic calculation (0*1 + 0*(-1) + 1*9) that does not follow from the problem statement or standard calculus theorems. The correct approach requires using the additive property of integrals: \int_0^6 g(x)dx = \int_0^1 g(x)dx + \int_1^6 g(x)dx, but the question asks for \int_1^0 g(x)dx, which is simply -\int_0^1 g(x)dx = -(-9) = 9. While the final number is correct, the reasoning provided in the solution is completely disconnected from the problem setup and mathematically unjustified.gpt-oss:20b: inconclusive 2026-10-06 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution’s second line misapplies the given data. It ignores the fact that \(\int_0^1 g(x)\,dx=-9\) and incorrectly combines unrelated terms (0*1, 0*(qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution ignores the given values for f(x) and incorrectly calculates the result as 9 instead of the correct value 10. It also fails to use the property that \int_1^0 g(x) dx = -\int_0^1 g(x) dx = -(-9) = 9, but the calculation shown (0*1 + 0*(-1) + 1*9) is nonsensical in this context and does not follow from the stated principles.
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 2026-10-06 with SymPy 1.14.0.