Changing the order of integration
Problem 11.187 · hard
Evaluate \( \displaystyle \int\limits_{0}^{1}\int\limits_{y}^{1} \sin{\left(x^{2} \right)}\, dx\, dy \) by reversing the order of integration.
- The inner integral has no elementary antiderivative in this order. The region is 0 ≤ y ≤ 1, y ≤ x ≤ 1; describe it the other way round.Reviewed
- The same region, integrated in the other order: Integral(sin(x**2), (y, 0, x), (x, 0, 1)). (The original order cannot be done by hand; the independent check below integrates it numerically.)Reviewed
- \[ \int\limits_{0}^{1}\int\limits_{0}^{x} \sin{\left(x^{2} \right)}\, dy\, dx = \frac{1}{2} - \frac{\cos{\left(1 \right)}}{2} \]Now the inner integral is easy, and the outer one is a substitution.✓ Proved
Answer \( \frac{1}{2} - \frac{\cos{\left(1 \right)}}{2} \approx 0.229849 \)
✓ Nihil obstat Lines: 1 proved, 2 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 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 | numerical quadrature in the original order |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the region, reverses the order of integration, and applies the correct limits and integrand. The final result is accurate.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the region, reverses the order of integration, and applies the correct limits and integrand. The final result is accurate.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the region of integration, reverses the order of integration to make the integral solvable, and arrives at the correct result.gpt-oss:20b: pass 2026-10-04
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/change_order, checked 2026-10-04 with SymPy 1.14.0.