Triple integrals over general regions
Problem 11.281 · medium
Evaluate \( \displaystyle \iiint_E x y\, dV \), where \( \displaystyle E = \{(x, y, z) : 0 \le x \le 2,\ 0 \le y \le x,\ 0 \le z \le x + y\} \).
- \[ \int\limits_{0}^{x + y} x y\, dz = x^{2} y + x y^{2} \]Integrate in z first.✓ Proved
- \[ \int\limits_{0}^{x} x y \left(x + y\right)\, dy = \frac{5 x^{4}}{6} \]Then in y.✓ Proved
- \[ \int\limits_{0}^{2} \frac{5 x^{4}}{6}\, dx = \frac{16}{3} \]Then in x.✓ Proved
Answer \( \frac{16}{3} \)
✓ 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 for the region E and performs the integration steps accurately.
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.