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Properties of definite integrals

Problem 4.330 · easy

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

The integral of a sum is the sum of the integrals, and constants factor out.
\[ \left(-1\right) 8 + 0 \cdot 3 + 2 \cdot 2 = -4 \]
Combine the known values.✓ Proved

Answer \( -4 \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution incorrectly uses the value of the integral from 0 to 4 for f(x) (which is 4) instead of the given value for the integral from 0 to 3 (which is 8) in the calculation, but then arbitrarily substitutes 8 in the equation line without justification or correct derivation. Wait, looking closer: The problem gives int_0^3 f(x) dx = 8. The solution uses 8. But it writes '-1*8'. This corresponds to -int_0^3 f(x) dx. Then it adds '0*3' which is meaningless noise. Then '2*2' corresponds to 2*int_0^3 g(x) dx. The result -4 is correct based on the inputs (-8 + 4 = -4). However, the sentence 'The integral of a sum is the sum of the integrals...' is a valid principle, but the equation line is presented as a black box calculation ' -1*8 + 0*3 + 2*2 = -4' without showing the integral properties explicitly applied to the specific integrals. More critically, the '0*3' term is completely unjustified and confusing, suggesting a misunderstanding of how to handle the bounds or a typo that obscures the logic. A student would not learn how to split the integral properly from this.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution incorrectly uses the value of the integral from 0 to 4 for f(x) (which is 4) instead of the given value for the integral from 0 to 3 (which is 8) in the calculation, but then arbitrarily substitutes 8 in the equation line without justification or correct derivation. Wait, looking closer: The problem gives int_0^3 f(x) dx = 8. The solution uses 8. But it writes '-1*8'. This corresponds to -int_0^3 f(x) dx. Then it adds '0*3' which is meaningless noise. Then '2*2' corresponds to 2*int_0^3 g(x) dx. The result -4 is correct based on the inputs (-8 + 4 = -4). However, the sentence 'The integral of a sum is the sum of the integrals...' is a valid principle, but the equation line is presented as a black box calculation ' -1*8 + 0*3 + 2*2 = -4' without showing the integral properties explicitly applied to the specific integrals. More critically, the '0*3' term is completely unjustified and confusing, suggesting a misunderstanding of how to handle the bounds or a typo that obscures the logic. A student would not learn how to split the integral properly from this.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution incorrectly substitutes the value of the integral from 0 to 4 (which is 4) with 0 for the f(x) term, instead of using the given value for the integral from 0 to 3 (which is 8). Specifically, it calculates -1*8 + 2*2 = -4, but the term '0*3' is nonsensical and suggests a confusion about which bounds correspond to which values, although the final arithmetic accidentally matches the correct result if one ignores the erroneous middle term. Wait, let's re-read carefully. The problem asks for integral from 0 to 3. Given: int_0^3 f = 8. So -1 * int_0^3 f = -1 * 8 = -8. Given: int_0^3 g = 2. So 2 * int_0^3 g = 2 * 2 = 4. Sum = -8 + 4 = -4. The solution writes '-1*8 + 0*3 + 2*2'. The '0*3' term is completely unjustified and incorrect notation. It seems to be trying to account for the bounds or perhaps a typo for something else, but it is mathematically nonsensical in this context. However, the final answer -4 is correct. The sentence 'Combine the known values' is vague. The equation line contains a garbage term '0*3'. This is a severe error in presentation/logic, even if the final number is right by coincidence or by ignoring the garbage term. A student would be confused by '0*3'. Is it an error? Yes. The step is not logically sound.
gpt-oss:20b: fail (error) 2026-10-04 — The solution misapplies the given integrals: it uses the value of ∫₀⁴g(x) instead of ∫₀³g(x) and includes an extraneous 0·3 term. The correct computation is -∫₀³f(x)+2∫₀³g(x)= -8+4 = -4.

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.