Triple integrals over general regions
Problem 11.284 · medium
Evaluate \( \displaystyle \iiint_E z\, dV \), where \( \displaystyle E = \{(x, y, z) : 0 \le x \le 3,\ 0 \le y \le 3 - x,\ 0 \le z \le - x - y + 3\} \).
- \[ \int\limits_{0}^{- x - y + 3} z\, dz = \frac{x^{2}}{2} + x y - 3 x + \frac{y^{2}}{2} - 3 y + \frac{9}{2} \]Integrate in z first.✓ Proved
- \[ \int\limits_{0}^{3 - x} \frac{\left(- x - y + 3\right)^{2}}{2}\, dy = - \frac{x^{3}}{6} + \frac{3 x^{2}}{2} - \frac{9 x}{2} + \frac{9}{2} \]Then in y.✓ Proved
- \[ \int\limits_{0}^{3} \left(\frac{\left(3 - x\right)^{3}}{6} + \left(3 - x\right)^{2} \left(\frac{x}{2} - \frac{3}{2}\right) + \left(3 - x\right) \left(\frac{x^{2}}{2} - 3 x + \frac{9}{2}\right)\right)\, dx = \frac{27}{8} \]Then in x.✓ Proved
Answer \( \frac{27}{8} \)
✓ 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 | nested numerical quadrature |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly sets up the iterated integral based on the region E and performs the integration steps accurately. The final result matches the stated answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly sets up the iterated integral based on the region E and performs the integration steps accurately. The final result matches the stated answer.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly sets up the iterated integral with the appropriate bounds and order of integration. The algebraic steps, although presented in a somewhat expanded form in the final step, are correct and lead to the right 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 2026-10-07 with SymPy 1.14.0.