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Triple integrals over general regions

Problem 11.348 · medium

Evaluate \( \displaystyle \iiint_E y\, dV \), where \( \displaystyle E = \{(x, y, z) : 0 \le x \le 3,\ 0 \le y \le 3 - x,\ 0 \le z \le - x - y + 3\} \).

\[ \int\limits_{0}^{- x - y + 3} y\, dz = - x y - y^{2} + 3 y \]
Integrate in z first.✓ Proved
\[ \int\limits_{0}^{3 - x} y \left(- x - y + 3\right)\, dy = - \frac{x^{3}}{6} + \frac{3 x^{2}}{2} - \frac{9 x}{2} + \frac{9}{2} \]
Then in y.✓ Proved
\[ \int\limits_{0}^{3} \left(\left(\frac{3}{2} - \frac{x}{2}\right) \left(3 - x\right)^{2} - \frac{\left(3 - x\right)^{3}}{3}\right)\, dx = \frac{27}{8} \]
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: pass
qwen3.6:27b-mlx: pass — The solution correctly sets up the iterated integral with the appropriate bounds for the region E and performs the integration in the correct order (z, then y, then x). The algebraic steps are verified as correct.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly sets up the iterated integral with the appropriate bounds for the region E and performs the integration in the correct order (z, then y, then x). The algebraic steps are verified as correct.
gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly sets up the iterated integral with the appropriate bounds and order of integration. The intermediate algebraic steps and final result are correct.

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.