Triple integrals over general regions
Problem 11.241 · medium
Evaluate \( \displaystyle \iiint_E x + 2 y z\, dV \), where \( \displaystyle E = \{(x, y, z) : 0 \le x \le 2,\ 0 \le y \le x,\ 0 \le z \le x + y\} \).
- \[ \int\limits_{0}^{x + y} \left(x + 2 y z\right)\, dz = x^{2} y + x^{2} + 2 x y^{2} + x y + y^{3} \]Integrate in z first.✓ Proved
- \[ \int\limits_{0}^{x} \left(x \left(x + y\right) + y \left(x + y\right)^{2}\right)\, dy = \frac{17 x^{4}}{12} + \frac{3 x^{3}}{2} \]Then in y.✓ Proved
- \[ \int\limits_{0}^{2} \left(\frac{11 x^{4}}{12} + x^{3} + x^{2} \left(\frac{x^{2}}{2} + \frac{x}{2}\right)\right)\, dx = \frac{226}{15} \]Then in x.✓ Proved
Answer \( \frac{226}{15} \)
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 appropriate bounds derived from the region E. The algebraic integration steps are verified as correct, leading to the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly sets up the iterated integral with the appropriate bounds derived from the region E. The algebraic integration steps are verified as correct, leading to the stated answer.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution contains no explanatory sentences, only unchecked algebraic steps. Specifically, the intermediate results in steps 1 and 2 are algebraically incorrect (e.g., the z-integral of x+2yz is x^2+xy+xy^2+y^3, not the expression shown), leading to an incorrect final answer of 226/15 instead of the correct 176/15.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.