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Definite integrals from area formulas

Problem 4.1367 · easy Video

Evaluate \( \displaystyle \int_{2}^{5} - x \, dx \) using a formula from geometry.

The region between the graph and the x-axis is a trapezoid, below the axis.
\[ \frac{\left(-5 - 2\right) \left(-2 + 5\right)}{2} = - \frac{21}{2} \]
Its (signed) area.✓ Proved
\[ \int\limits_{2}^{5} \left(- x\right)\, dx = - \frac{21}{2} \]
Integrating directly gives the same.✓ Proved

Answer \( - \frac{21}{2} \)

The graph of f(x) = -x from x = 2 to x = 5. Rectangles under the curve get thinner until they fill the area, -21/2. 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: fail (misleading) — The sentence correctly identifies the region as a trapezoid, but the area computation in line 2 uses an incorrect algebraic expression for the bases: it writes (-5‑2) instead of the correct difference (-5)‑(-2)=‑3. Although the final numeric result is correct, the erroneous formula could mislead a student about how to apply the trapezoid area formula.
qwen3.6:27b-mlx: pass — The solution correctly identifies the geometric shape as a trapezoid and applies the area formula with the correct sign for the region below the x-axis. The arithmetic is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-29 — The solution correctly identifies the geometric shape as a trapezoid and applies the area formula with the correct sign for the region below the x-axis. The arithmetic is correct.
gpt-oss:20b: fail (misleading) 2026-10-29 — The sentence correctly identifies the region as a trapezoid, but the area computation in line 2 uses an incorrect algebraic expression for the bases: it writes (-5‑2) instead of the correct difference (-5)‑(-2)=‑3. Although the final numeric result is correct, the erroneous formula could mislead a student about how to apply the trapezoid area formula.
qwen3.6:27b-mlx: fail (error) 2026-10-28 — The solution incorrectly identifies the region as a trapezoid; the region under y = -x from 2 to 5 is a trapezoid only if bounded by vertical lines, but the description is vague and the calculation in line 2 uses a formula for the area of a trapezoid with heights 2 and 5 and width 3, which is (2+5)*3/2 = 10.5, but applies a negative sign manually. While the numerical result is correct, the geometric reasoning is flawed because it doesn't explicitly define the parallel sides and height correctly in standard terms, and more importantly, line 2's expression `(-5 - 2)*(-2 + 5)/2` is algebraically confusing and doesn't clearly map to the standard trapezoid area formula `(h1+h2)/2 * w`. Specifically, `(-5-2)` is -7 and `(-2+5)` is 3, so `-7*3/2 = -10.5`. This looks like it's trying to use `(y2 - y1)` or similar, but the standard geometric area is positive, and the signed area is negative. The sentence says 'below the axis', implying signed area is negative. The calculation `(-5 - 2)` suggests using y-values? No, y(2)=-2, y(5)=-5. The average height is (-2-5)/2 = -3.5. Width is 3. Area = -10.5. The formula used `(-5-2)` is `y(5) - y(2)`? No, that's -3. It's just a messy calculation that happens to work. However, the bigger issue is that calling it a 'trapezoid' is correct, but the explanation is poor. Actually, looking closer, the prompt asks to evaluate using a formula from geometry. The solution does this. Is it wrong? The calculation `(-5 - 2)*(-2 + 5)/2` is `-7 * 3 / 2 = -10.5`. This is not the standard trapezoid area formula `(a+b)/2 * h`. Here a=2, b=5 (lengths), h=3. Area = 10.5. Signed area = -10.5. The calculation in line 2 is not the standard geometric formula, it's a weird algebraic manipulation that yields the right number. It's misleading because it doesn't teach the geometric formula. It teaches a specific algebraic trick that might not generalize. But is it an 'error'? The verdict 'fail' with 'misleading' or 'error' is appropriate. Let's call it 'misleading' because the student learns a non-standard, confusing way to calculate the area rather than the geometric formula.
gpt-oss:20b: fail (error) 2026-10-28 — The trapezoid area formula is applied incorrectly: the lengths of the parallel sides should be 2 and 5, not –5 and –2. Using (–5–2)(–2+5)/2 yields a negative value, whereas the signed area of the region is –21/2, but the geometric area is +21/2. The solution mixes signed and unsigned areas in a confusing way.
qwen3.6:27b-mlx: inconclusive 2026-10-28 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "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, x=2, and x=5 is a trapezoid o
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.