Triple integrals over general regions
Problem 11.286 · medium
Evaluate \( \displaystyle \iiint_E x + y + z\, dV \), where \( \displaystyle E = \{(x, y, z) : 0 \le x \le 1,\ 0 \le y \le \sqrt{x},\ 0 \le z \le y\} \).
- \[ \int\limits_{0}^{y} \left(x + y + z\right)\, dz = x y + \frac{3 y^{2}}{2} \]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
- \[ \int\limits_{0}^{1} \left(\frac{x^{\frac{3}{2}}}{2} + \frac{x^{2}}{2}\right)\, dx = \frac{11}{30} \]Then in x.✓ Proved
Answer \( \frac{11}{30} \)
✓ 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
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07gpt-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 marked unchecked, are consistent with the stated answer and standard integration techniques.
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.