Trapezoidal, midpoint and Simpson's rules
Problem 4.975 · medium
Approximate \( \displaystyle \int_{0}^{2} x^{3} + 1\, dx \) using the midpoint rule with \( \displaystyle n = 4 \).
- \[ 1 \cdot \frac{1}{2} = \frac{1}{2} \]Δx = (b − a)/n.✓ Proved
- The midpoint rule uses the points x = 1/4, 3/4, 5/4, 7/4 with weights 1, 1, 1, 1, all times 1/2.Reviewed
- \[ \frac{1 \cdot 65 \cdot \frac{1}{64} + 1 \cdot 91 \cdot \frac{1}{64} + 1 \cdot 189 \cdot \frac{1}{64} + 1 \cdot 407 \cdot \frac{1}{64}}{2} = \frac{47}{8} \]Weighted sum of the function values.✓ Proved
- \[ \int\limits_{0}^{2} \left(x^{3} + 1\right)\, dx = 6 \]The exact value, ≈ 6.000000, for comparison: the error is about 1.25e-01.✓ Proved
Answer \( M_{4} = \frac{47}{8} \approx 5.875000 \)
✓ Nihil obstat Lines: 3 proved, 1 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 |
| 4 | ✓ 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 | the rule implemented separately in floating point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the subinterval width and midpoints, and the final calculation is accurate.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly identifies the subinterval width and midpoints, and the final calculation is accurate.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly identifies the subinterval width and midpoints, and the arithmetic for the midpoint approximation is 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/numerical_integration, checked 2026-10-12 with SymPy 1.14.0.