Triple integrals over general regions
Problem 11.234 · medium
Evaluate \( \displaystyle \iiint_E x z\, dV \), where \( \displaystyle E = \{(x, y, z) : -1 \le x \le 1,\ 0 \le y \le 1,\ 0 \le z \le 1 - x^{2}\} \).
- \[ \int\limits_{0}^{1 - x^{2}} x z\, dz = \frac{x^{5}}{2} - x^{3} + \frac{x}{2} \]Integrate in z first.✓ Proved
- \[ \int\limits_{0}^{1} \frac{x \left(1 - x^{2}\right)^{2}}{2}\, dy = \frac{x^{5}}{2} - x^{3} + \frac{x}{2} \]Then in y.✓ Proved
- \[ \int\limits_{-1}^{1} \frac{x \left(1 - x^{2}\right)^{2}}{2}\, dx = 0 \]Then in x.✓ Proved
Answer \( 0 \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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: inconclusive — reviewer returned a non-object
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer returned a non-objectgpt-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.