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Properties of definite integrals

Problem 4.494 · easy

Suppose \( \displaystyle \int_0^{4} f(x)\,dx = -3 \), \( \displaystyle \int_0^{1} f(x)\,dx = -2 \), \( \displaystyle \int_0^{4} g(x)\,dx = -7 \) and \( \displaystyle \int_0^{1} g(x)\,dx = -8 \). Find \( \displaystyle \int_{1}^{4} \left(f(x)\right) dx \).

The integral of a sum is the sum of the integrals, and constants factor out; ∫ from 1 to 4 = ∫ from 0 to 4 − ∫ from 0 to 1.
\[ 1 \left(-1\right) + 0 \cdot 1 + 0 \cdot 3 = -1 \]
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. The reviewers disagree about how one step is explained; every verdict is in 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 second sentence incorrectly combines the known values; it should use \(-3 - (-2) = -1\) rather than the nonsensical expression shown."}
qwen3.6:27b-mlx: fail (error) — The solution ignores the provided data for g(x) and incorrectly calculates the result as -1 instead of -1. While the final number matches the stated answer, the reasoning is flawed because it treats the problem as if only f(x) matters (which is true for the specific question asked, but the solution text implies a generic 'sum' rule application that is confusingly written). More critically, the equation line '1*(-1) + 0*1 + 0*3 = -1' is nonsensical in the context of the given integrals. The correct calculation is simply \int_1^4 f(x) dx = \int_0^4 f(x) dx - \int_0^1 f(x) dx = -3 - (-2) = -1. The provided equation does not reflect this subtraction clearly and introduces arbitrary coefficients.

Every verdict on record (4)

gpt-oss:20b: inconclusive 2026-10-06 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The second sentence incorrectly combines the known values; it should use \(-3 - (-2) = -1\) rather than the nonsensical expression shown."}
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution ignores the provided data for g(x) and incorrectly calculates the result as -1 instead of -1. While the final number matches the stated answer, the reasoning is flawed because it treats the problem as if only f(x) matters (which is true for the specific question asked, but the solution text implies a generic 'sum' rule application that is confusingly written). More critically, the equation line '1*(-1) + 0*1 + 0*3 = -1' is nonsensical in the context of the given integrals. The correct calculation is simply \int_1^4 f(x) dx = \int_0^4 f(x) dx - \int_0^1 f(x) dx = -3 - (-2) = -1. The provided equation does not reflect this subtraction clearly and introduces arbitrary coefficients.
gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution ignores the provided information about g(x) and incorrectly computes the result as -1 instead of the correct value -1 (wait, let me re-calculate: -3 - (-2) = -1. The answer is actually correct, but the reasoning is flawed because it mentions 'integral of a sum' which is irrelevant, and the equation line is nonsensical garbage '1*(-1) + 0*1 + 0*3'. The sentence claims 'integral of a sum' but the problem doesn't involve a sum. The equation is completely hallucinated. The final answer happens to be correct by chance or separate calculation, but the provided steps are wrong.)

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.