Triple integrals
Problem 11.123 · easy
Evaluate \( \displaystyle \iiint_B x^{2} z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 3] \times [0, 3] \).
- \[ \int\limits_{0}^{3} x^{2} z\, dz = \frac{9 x^{2}}{2} \]Integrate in z.✓ Proved
- \[ \int\limits_{0}^{3} \frac{9 x^{2}}{2}\, dy = \frac{27 x^{2}}{2} \]Then y.✓ Proved
- \[ \int\limits_{0}^{2} \frac{27 x^{2}}{2}\, dx = 36 \]Then x.✓ Proved
Answer \( 36 \)
✓ 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 the iterated integral over the rectangular box B and performs the integration in the correct order with correct bounds and results.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly sets up the iterated integral over the rectangular box B and performs the integration in the correct order with correct bounds and results.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly sets up the iterated integral over the rectangular box B and performs the integration steps accurately. The final result matches the stated answer.gpt-oss:20b: pass 2026-09-26
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 2026-09-26 with SymPy 1.14.0.