Triple integrals over general regions
Problem 11.351 · medium
Evaluate \( \displaystyle \iiint_E x + 1\, dV \), where \( \displaystyle E = \{(x, y, z) : 0 \le x \le 1,\ 0 \le y \le x,\ 0 \le z \le x + y\} \).
- \[ \int\limits_{0}^{x + y} \left(x + 1\right)\, dz = x^{2} + x y + x + y \]Integrate in z first.✓ Proved
- \[ \int\limits_{0}^{x} \left(x + 1\right) \left(x + y\right)\, dy = \frac{3 x^{3}}{2} + \frac{3 x^{2}}{2} \]Then in y.✓ Proved
- \[ \int\limits_{0}^{1} \left(x^{2} \left(\frac{x}{2} + \frac{1}{2}\right) + x \left(x^{2} + x\right)\right)\, dx = \frac{7}{8} \]Then in x.✓ Proved
Answer \( \frac{7}{8} \)
Every line of this solution was proved by the computer algebra system SymPy. 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ 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 | nested numerical quadrature |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly sets up the iterated integral with the proper bounds and order of integration, and the algebraic steps are verified as correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly sets up the iterated integral with the proper bounds and order of integration, and the algebraic steps are verified as correct.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (error) 2026-10-10 — The integration steps are mathematically incorrect. Specifically, the first integration with respect to z yields x^2 + xy + x + y, but the second step incorrectly treats the integrand as (x+1)(x+y) instead of the correct result from step 1, leading to an incorrect final answer.
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/triple_integral_region, checked 2026-10-10 with SymPy 1.14.0.