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Definite integrals from area formulas

Problem 4.1370 · easy Video

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

The region between the graph and the x-axis is a trapezoid, below the axis.
\[ \frac{\left(-4 - 1\right) \left(-1 + 4\right)}{2} = - \frac{15}{2} \]
Its (signed) area.✓ Proved
\[ \int\limits_{1}^{4} \left(- x\right)\, dx = - \frac{15}{2} \]
Integrating directly gives the same.✓ Proved

Answer \( - \frac{15}{2} \)

The graph of f(x) = -x from x = 1 to x = 4. Rectangles under the curve get thinner until they fill the area, -15/2. 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 claims the region is a trapezoid, but the graph of y = -x is a line, so the region bounded by the x-axis and the vertical lines x=1 and x=4 is a triangle, not a trapezoid. While the numerical result is correct, the geometric identification is factually wrong.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-29 — The solution claims the region is a trapezoid, but the graph of y = -x is a line, so the region bounded by the x-axis and the vertical lines x=1 and x=4 is a triangle, not a trapezoid. While the numerical result is correct, the geometric identification is factually wrong.
gpt-oss:20b: pass 2026-10-29
qwen3.6:27b-mlx: fail (error) 2026-10-28 — The solution incorrectly identifies the region as a trapezoid; the graph of y = -x is a straight line, so the region bounded by the x-axis and the vertical lines x=1 and x=4 is a triangle, not a trapezoid. While the numerical result is correct, the geometric classification is factually wrong.
gpt-oss:20b: pass 2026-10-28
qwen3.6:27b-mlx: fail (error) 2026-10-28 — The geometric formula for the area of a trapezoid yields a positive value (7.5). The solution incorrectly applies a negative sign to the geometric area formula itself, rather than correctly stating that the integral equals the negative of the area. This conflates geometric area (always non-negative) with signed area, teaching a false method.
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.