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Properties of definite integrals

Problem 4.500 · easy

Suppose \( \displaystyle \int_0^{2} f(x)\,dx = -6 \), \( \displaystyle \int_0^{1} f(x)\,dx = 4 \), \( \displaystyle \int_0^{2} g(x)\,dx = -8 \) and \( \displaystyle \int_0^{1} g(x)\,dx = -3 \). Find \( \displaystyle \int_{1}^{0} \left(- f(x) + 2 g(x)\right) dx \).

The integral of a sum is the sum of the integrals, and constants factor out; swapping the limits changes the sign.
\[ 0 \left(-1\right) - 1 \left(-4\right) + 2 \cdot 3 = 10 \]
Combine the known values.✓ Proved

Answer \( 10 \)

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: fail (misleading) — The second sentence uses incorrect values (0*(-1) and 2*3) instead of the correct integrals – a student could be misled about how the numbers are combined, even though the final answer happens to be right.
qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object

Every verdict on record (4)

gpt-oss:20b: fail (misleading) 2026-10-06 — The second sentence uses incorrect values (0*(-1) and 2*3) instead of the correct integrals – a student could be misled about how the numbers are combined, even though the final answer happens to be right.
qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to explicitly calculate the necessary integral values (e.g., \int_1^2 f(x)dx) from the given data, jumping directly to a numerical combination that lacks clear derivation. Specifically, it is unclear how the terms -1, -4, and 3 relate to the given integrals without showing the intermediate step of splitting the integral from 0 to 2 into 0 to 1 and 1 to 2.

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.