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Triple integrals over general regions

Problem 11.319 · medium

Evaluate \( \displaystyle \iiint_E z\, dV \), where \( \displaystyle E = \{(x, y, z) : -1 \le x \le 1,\ 0 \le y \le 1,\ 0 \le z \le 1 - x^{2}\} \).

\[ \int\limits_{0}^{1 - x^{2}} z\, dz = \frac{x^{4}}{2} - x^{2} + \frac{1}{2} \]
Integrate in z first.✓ Proved
\[ \int\limits_{0}^{1} \frac{\left(1 - x^{2}\right)^{2}}{2}\, dy = \frac{x^{4}}{2} - x^{2} + \frac{1}{2} \]
Then in y.✓ Proved
\[ \int\limits_{-1}^{1} \frac{\left(1 - x^{2}\right)^{2}}{2}\, dx = \frac{8}{15} \]
Then in x.✓ Proved

Answer \( \frac{8}{15} \)

✓ 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 proper bounds and order of integration, and the algebraic steps are verified as correct.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly sets up the iterated integral with the proper bounds and order of integration, and the algebraic steps are verified as correct.
gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly sets up the iterated integral with the proper bounds and order of integration. The algebraic steps are verified as correct, leading to the right final 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 2026-10-08 with SymPy 1.14.0.