Triple integrals over general regions
Problem 11.285 · medium
Evaluate \( \displaystyle \iiint_E z\, dV \), where \( \displaystyle E = \{(x, y, z) : -1 \le x \le 1,\ 0 \le y \le 2,\ 0 \le z \le 1 - x^{2}\} \).
- \[ \int\limits_{0}^{1 - x^{2}} z\, dz = \frac{x^{4}}{2} - x^{2} + \frac{1}{2} \]Integrate in z first.✓ Proved
- \[ \int\limits_{0}^{2} \frac{\left(1 - x^{2}\right)^{2}}{2}\, dy = x^{4} - 2 x^{2} + 1 \]Then in y.✓ Proved
- \[ \int\limits_{-1}^{1} \left(1 - x^{2}\right)^{2}\, dx = \frac{16}{15} \]Then in x.✓ Proved
Answer \( \frac{16}{15} \)
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: fail (error) — The y‑integration step is incorrect: after integrating over z the integrand is (1−x²)²/2, but multiplying by the y‑interval length 2 gives (1−x²)², not (1−x²)²/2. This propagates a factor‑of‑2 error, leading to the wrong final value 16/15 instead of the correct 32/15.qwen3.6:27b-mlx: pass — The solution correctly sets up the iterated integral with the appropriate bounds and order of integration. The intermediate steps and final result are algebraically correct.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-07 — The y‑integration step is incorrect: after integrating over z the integrand is (1−x²)²/2, but multiplying by the y‑interval length 2 gives (1−x²)², not (1−x²)²/2. This propagates a factor‑of‑2 error, leading to the wrong final value 16/15 instead of the correct 32/15.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly sets up the iterated integral with the appropriate bounds and order of integration. The intermediate steps and final result are algebraically correct.gpt-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 and final result are correct.
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.