Home›Calculus 1›Definite integrals from area formulas›Problem 4.1375

Definite integrals from area formulas

Problem 4.1375 · easy Video

Evaluate \( \displaystyle \int_{2}^{3} - 3 x - 1 \, dx \) using a formula from geometry.

The region between the graph and the x-axis is a trapezoid, below the axis.
\[ \frac{\left(-10 - 7\right) \left(-2 + 3\right)}{2} = - \frac{17}{2} \]
Its (signed) area.✓ Proved
\[ \int\limits_{2}^{3} \left(- 3 x - 1\right)\, dx = - \frac{17}{2} \]
Integrating directly gives the same.✓ Proved

Answer \( - \frac{17}{2} \)

The graph of f(x) = -3x - 1 from x = 2 to x = 3. Rectangles under the curve get thinner until they fill the area, -17/2. Area below the x-axis counts as negative.

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: fail (error) — The solution incorrectly labels the region as a trapezoid; the function is linear, so the region is a trapezoid, but the calculation in line 2 uses the formula for the area of a trapezoid incorrectly by mixing signed values without absolute values for geometric area, then applying a sign. More critically, the geometric area should be calculated as positive (area = 17/2) and then the sign applied because it is below the axis. The calculation (-10 - 7)*(-2 + 3)/2 is algebraically -17/2, but geometrically, one computes the area of the trapezoid with heights 7 and 10 (absolute values) and width 1, getting 17/2, then assigns the negative sign. While the final number is correct, the phrasing 'Its (signed) area' applied to a calculation that subtracts negative numbers directly is confusing and technically misapplies the geometric formula which requires positive lengths. A clearer geometric approach would calculate the area of the trapezoid with bases 7 and 10 and height 1, yielding 17/2, and then state the integral is -17/2 because the region is below the axis.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-29 — The solution incorrectly labels the region as a trapezoid; the function is linear, so the region is a trapezoid, but the calculation in line 2 uses the formula for the area of a trapezoid incorrectly by mixing signed values without absolute values for geometric area, then applying a sign. More critically, the geometric area should be calculated as positive (area = 17/2) and then the sign applied because it is below the axis. The calculation (-10 - 7)*(-2 + 3)/2 is algebraically -17/2, but geometrically, one computes the area of the trapezoid with heights 7 and 10 (absolute values) and width 1, getting 17/2, then assigns the negative sign. While the final number is correct, the phrasing 'Its (signed) area' applied to a calculation that subtracts negative numbers directly is confusing and technically misapplies the geometric formula which requires positive lengths. A clearer geometric approach would calculate the area of the trapezoid with bases 7 and 10 and height 1, yielding 17/2, and then state the integral is -17/2 because the region is below the axis.
gpt-oss:20b: pass 2026-10-29
qwen3.6:27b-mlx: pass 2026-10-28 — The solution correctly identifies the region as a trapezoid below the x-axis and applies the geometric area formula with the correct sign to match the definite integral.
gpt-oss:20b: pass 2026-10-28
qwen3.6:27b-mlx: inconclusive 2026-10-28 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly applies the trapezoid area formula by using negative function values directly, resulting in a positive geometric area (-17/
gpt-oss:20b: pass 2026-10-28

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 2026-10-29 with SymPy 1.14.0.