Home›Calculus 1›Definite integrals from area formulas›Problem 4.1149

Definite integrals from area formulas

Problem 4.1149 · easy Video

Evaluate \( \displaystyle \int_{-1}^{4} 3 x + 4 \, dx \) using a formula from geometry.

The region between the graph and the x-axis is a trapezoid.
\[ \frac{\left(1 + 4\right) \left(1 + 16\right)}{2} = \frac{85}{2} \]
Its (signed) area.✓ Proved
\[ \int\limits_{-1}^{4} \left(3 x + 4\right)\, dx = \frac{85}{2} \]
Integrating directly gives the same.✓ Proved

Answer \( \frac{85}{2} \)

The graph of f(x) = 3x + 4 from x = -1 to x = 4. Rectangles under the curve get thinner until they fill the area, 85/2.

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: pass
qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution incorrectly identifies the region as a trapezoid and applies the trapezoid area formula to the x-interval length and function values, ignoring that the function crosses the x-axis at x = -4/3. This creates two triangles (one with negative signed area) rather than a single trapezoid, leading to an incorrect geometric interpretation even though the final numerical answer coincidentally matches the integral due to a calculation error in the 'proved' equation (the actual integral is 37.5, not 42.5).
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer returned a non-object
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 2026-10-18 with SymPy 1.14.0.