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Triple integrals over general regions

Problem 11.235 · medium

Evaluate \( \displaystyle \iiint_E x + y + z\, dV \), where \( \displaystyle E = \{(x, y, z) : 0 \le x \le 4,\ 0 \le y \le \sqrt{x},\ 0 \le z \le y\} \).

\[ \int\limits_{0}^{y} \left(x + y + z\right)\, dz = x y + \frac{3 y^{2}}{2} \]
Integrate in z first.✓ Proved
\[ \int\limits_{0}^{\sqrt{x}} \left(\frac{y^{2}}{2} + y \left(x + y\right)\right)\, dy = \frac{x^{\frac{3}{2}}}{2} + \frac{x^{2}}{2} \]
Then in y.✓ Proved
\[ \int\limits_{0}^{4} \left(\frac{x^{\frac{3}{2}}}{2} + \frac{x^{2}}{2}\right)\, dx = \frac{256}{15} \]
Then in x.✓ Proved

Answer \( \frac{256}{15} \)

✓ 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)

qwen3.6:27b-mlx: pass 2026-10-05 — 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 2026-10-05
qwen3.6:27b-mlx: pass 2026-10-05
gpt-oss:20b: pass 2026-10-05

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 2026-10-05 with SymPy 1.14.0.