Triple integrals
Problem 11.146 · easy
Evaluate \( \displaystyle \iiint_B x y z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 2] \times [0, 3] \).
- \[ \int\limits_{0}^{3} x y z\, dz = \frac{9 x y}{2} \]Integrate in z.✓ Proved
- \[ \int\limits_{0}^{2} \frac{9 x y}{2}\, dy = 9 x \]Then y.✓ Proved
- \[ \int\limits_{0}^{2} 9 x\, dx = 18 \]Then x.✓ Proved
Answer \( 18 \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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 | a 40×40×40 midpoint sum over the box agrees to 1% |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly sets up and evaluates the iterated integral over the rectangular box B. The algebraic steps are verified, and the final result is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly sets up and evaluates the iterated integral over the rectangular box B. The algebraic steps are verified, and the final result is correct.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly sets up the iterated integral with the appropriate bounds and computes the result accurately.gpt-oss:20b: pass 2026-09-27
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_box, checked 2026-09-27 with SymPy 1.14.0.