Riemann sums: left, right and midpoint
Problem 4.855 · medium
Compute the left Riemann sum \( \displaystyle L_{5} \) for \( \displaystyle f(x) = 2 x^{3} + 1 \) on \( \displaystyle [-1, 1] \).
- \[ \frac{2}{5} \]Δx = (b − a)/n.✓ Proved
- The left endpoints are x = -1, -3/5, -1/5, 1/5, 3/5.
- \[ \frac{2 \left(-1 + \frac{71}{125} + \frac{123}{125} + \frac{127}{125} + \frac{179}{125}\right)}{5} = \frac{6}{5} \]Δx times the sum of the function values there.✓ Proved
- For comparison, the exact integral is 2 ≈ 2.0000.
Answer \( L_{5} = \frac{6}{5} \approx 1.2000 \)
Lines: 2 proved, 2 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 | Not checked | — | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the sum recomputed in 30-digit floating point |
Reviewers
gpt-oss:20b: fail (error) — The computed left Riemann sum in step 3 is incorrect. Using Δx=2/5 and the function values at the left endpoints gives (2/5)*(−1+71/125+123/125+127/125+179/125)=1.424, not 6/5. The solution’s algebraic result is therefore wrong.qwen3.6:27b-mlx: fail (error) — The solution claims the exact integral is 2, but the integral of 2x^3 + 1 from -1 to 1 is actually 2. While the final numerical value is coincidentally correct because the odd part integrates to 0, the phrasing 'For comparison' implies the Riemann sum is an approximation of this value. However, the primary error is in line 3's intermediate calculation description: it lists terms like 71/125 which do not correspond to f(x) at the stated endpoints (e.g., f(-3/5) = 2(-27/125)+1 = 98/125, not 71/125). The arithmetic in line 3 is internally inconsistent with the function values at the stated points, even if the final sum happens to be correct due to a likely typo in the displayed intermediate terms vs the actual calculation performed.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-10 — The computed left Riemann sum in step 3 is incorrect. Using Δx=2/5 and the function values at the left endpoints gives (2/5)*(−1+71/125+123/125+127/125+179/125)=1.424, not 6/5. The solution’s algebraic result is therefore wrong.qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution claims the exact integral is 2, but the integral of 2x^3 + 1 from -1 to 1 is actually 2. While the final numerical value is coincidentally correct because the odd part integrates to 0, the phrasing 'For comparison' implies the Riemann sum is an approximation of this value. However, the primary error is in line 3's intermediate calculation description: it lists terms like 71/125 which do not correspond to f(x) at the stated endpoints (e.g., f(-3/5) = 2(-27/125)+1 = 98/125, not 71/125). The arithmetic in line 3 is internally inconsistent with the function values at the stated points, even if the final sum happens to be correct due to a likely typo in the displayed intermediate terms vs the actual calculation performed.qwen3.6:27b-mlx: inconclusive 2026-10-10 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-10
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/riemann_sums, checked 2026-10-10 with SymPy 1.14.0.