Home›Calculus 2›Trapezoidal, midpoint and Simpson's rules›Problem 4.386

Trapezoidal, midpoint and Simpson's rules

Problem 4.386 · medium

Approximate \( \displaystyle \int_{0}^{2} x^{2}\, dx \) using the trapezoidal rule with \( \displaystyle n = 4 \).

\[ 1 \cdot \frac{1}{2} = \frac{1}{2} \]
Δx = (b − a)/n.✓ Proved
The trapezoidal rule uses the points x = 0, 1/2, 1, 3/2, 2 with weights 1, 2, 2, 2, 1, all times 1/4.
\[ \frac{1 \cdot 0 + 2 \cdot 1 \cdot \frac{1}{4} + 2 \cdot 1 + 1 \cdot 4 + 2 \cdot 9 \cdot \frac{1}{4}}{4} = \frac{11}{4} \]
Weighted sum of the function values.✓ Proved
\[ \int\limits_{0}^{2} x^{2}\, dx = \frac{8}{3} \]
The exact value, ≈ 2.666667, for comparison: the error is about 8.33e-02.✓ Proved

Answer \( T_{4} = \frac{11}{4} \approx 2.750000 \)

Lines: 3 proved, 1 not checked. 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	Not checked	—	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: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The sentence in step 2 incorrectly states that the weights are multiplied by 1/4. The trapezoidal rule formula is (Δx/2) * [f(x_0) + 2f(x_1) + ... +

Every verdict on record (4)

qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The sentence in step 2 incorrectly states that the weights are multiplied by 1/4. The trapezoidal rule formula is (Δx/2) * [f(x_0) + 2f(x_1) + ... +
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly applies the trapezoidal rule formula. The standard formula is (Δx/2) * [f(x_0) + 2f(x_1) + ... + f(x_n)]. With Δx = 1/2, th
gpt-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/numerical_integration, checked 2026-10-05 with SymPy 1.14.0.