Definite integrals from area formulas
Problem 4.1071 · easy Video
Evaluate \( \displaystyle \int_{-1}^{2} 3 x + 4 \, dx \) using a formula from geometry.
- The region between the graph and the x-axis is a trapezoid.
- \[ \frac{\left(1 + 2\right) \left(1 + 10\right)}{2} = \frac{33}{2} \]Its (signed) area.✓ Proved
- \[ \int\limits_{-1}^{2} \left(3 x + 4\right)\, dx = \frac{33}{2} \]Integrating directly gives the same.✓ Proved
Answer \( \frac{33}{2} \)
Lines: 2 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. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 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 | numerical quadrature |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-17 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the region as a trapezoid and applies the geometric area formula with the correct base width (3) and average height (5.5) to arrive at the correct result.gpt-oss:20b: pass 2026-10-14
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/integral_geometry, checked 2026-10-17 with SymPy 1.14.0.