Properties of definite integrals
Problem 4.782 · easy
Suppose \( \displaystyle \int_0^{3} f(x)\,dx = 4 \), \( \displaystyle \int_0^{2} f(x)\,dx = 1 \), \( \displaystyle \int_0^{3} g(x)\,dx = 3 \) and \( \displaystyle \int_0^{2} g(x)\,dx = -8 \). Find \( \displaystyle \int_{0}^{2} \left(f(x) + g(x)\right) dx \).
- The integral of a sum is the sum of the integrals, and constants factor out.
- \[ 1 \left(-8\right) + 0 \cdot 2 + 1 \cdot 1 = -7 \]Combine the known values.✓ Proved
Answer \( -7 \)
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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The second line incorrectly combines the known integrals; it should simply add \int_0^2 f = 1 and \int_0^2 g = -8 to obtain -7. The expression 1*(-8)+0*2+qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
gpt-oss:20b: inconclusive 2026-10-09 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The second line incorrectly combines the known integrals; it should simply add \int_0^2 f = 1 and \int_0^2 g = -8 to obtain -7. The expression 1*(-8)+0*2+qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer returned a non-objectgpt-oss:20b: fail (error) 2026-10-09 — The solution’s second line misapplies the linearity of the integral. It incorrectly mixes constants and bounds (1*(-8)+0*2+1*1) instead of simply adding the known integrals 1 and -8 to obtain -7. The reasoning is flawed and would mislead a student.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to explicitly apply the linearity property to split the integral into two separate terms, jumping directly to a numerical calculation without showing the intermediate step of identifying the correct definite integrals. This makes the logic opaque and skips the crucial modeling step of matching the given bounds to the problem statement.
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 2026-10-09 with SymPy 1.14.0.