Definite integrals from area formulas
Problem 4.1068 · easy Video
Evaluate \( \displaystyle \int_{0}^{1} - 2 x - 1 \, dx \) using a formula from geometry.
- The region between the graph and the x-axis is a trapezoid, below the axis.Reviewed
- \[ \frac{\left(-3 - 1\right) \left(0 + 1\right)}{2} = -2 \]Its (signed) area.✓ Proved
- \[ \int\limits_{0}^{1} \left(- 2 x - 1\right)\, dx = -2 \]Integrating directly gives the same.✓ Proved
Answer \( -2 \)
Lines: 2 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the region as a trapezoid and applies the geometric area formula with the correct sign for the region below the x-axis.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the region as a trapezoid and applies the geometric area formula with the correct sign for the region below the x-axis.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The solution incorrectly identifies the region as a trapezoid; the function f(x) = -2x - 1 is linear, so the region is a trapezoid, but the calculation in step 2 uses the formula for the area of a trapezoid with heights f(0)=-1 and f(1)=-3. The formula used is (h1 + h2) * w / 2. Here h1 = -1, h2 = -3, w = 1. (-1 + -3)*1/2 = -2. The equation shows (-3 - 1)*(0 + 1)/2. This is mathematically correct for the signed area. However, the prompt asks to evaluate using a formula from geometry. Usually, this implies calculating the geometric area (positive) and applying the sign based on position. The solution calculates signed area directly. Is this wrong? No, signed area is a geometric concept. But wait, step 1 says 'The region ... is a trapezoid'. Step 2 calculates the signed area. Step 3 confirms. The main issue is likely that 'formula from geometry' usually refers to Area = 1/2 * (b1 + b2) * h. The solution applies this correctly to signed values. Let's look closer. Is there a subtle error? The problem is simple. Maybe the 'error' is that it's not a trapezoid? A region bounded by a line and the x-axis is a trapezoid (or triangle). Here it is a trapezoid. The calculation is correct. Why would it fail? Perhaps the phrasing 'Its (signed) area' is considered misleading if the student is expected to compute positive area and negate? Or is the calculation in line 2 actually wrong? (-3 - 1) = -4. (0+1) = 1. -4 * 1 / 2 = -2. Correct. Let's re-read the prompt constraints. 'Check that... the setup models the problem... right formula...'. The formula for the area of a trapezoid is A = (a+b)/2 * h. The solution uses (-3-1)/2 * 1. This is (f(1)+f(0))/2 * (1-0). This is the correct application of the trapezoidal rule/geometry for signed area. Is it possible the system considers 'geometry' to strictly mean positive area? If so, the student should calculate Area = 2 and then say Integral = -2. The solution skips the positive area step. This might be 'misleading' or 'style'. But is it an 'error'? In many calculus contexts, 'geometry' implies visualizing the shape. The shape is below the axis. The signed area is negative. The calculation is robust. Let's look for a harder error. Did the student identify the vertices correctly? (0, -1) and (1, -3). Yes. Is it a trapezoid? Yes. Is the formula applied correctly? Yes. Maybe the issue is that the problem asks for 'a formula from geometry' and the solution just does arithmetic without explicitly stating the geometric formula? No, step 1 identifies the shape. Step 2 applies the formula. I will mark it as pass unless I find a specific flaw. Wait, look at the equation in line 2: `(-3 - 1)*(0 + 1)/2 = -2`. The term `(0 + 1)` represents the width `1 - 0`. The term `(-3 - 1)` represents `f(1) + f(0)`. This is correct. However, standard geometric area formulas yield positive values. Using them with negative inputs to get signed area is an extension, not the standard 'formula from geometry' taught in introductory calculus (which usually says 'Area of trapezoid is...' and then 'Since it's below, Integral = -Area'). By conflating the two, the solution might be considered 'misleading' about what 'geometry formulas' are. But is it an error? I'll lean towards 'misleading' or 'pass'. Let's look at the provided solution again. It says 'Its (signed) area'. This acknowledges the sign. I will give it a pass. Wait, I must be adversarial. Is there any ambiguity? No. Is the answer -2? Yes. Is the method geometric? Yes. I will pass it. Actually, looking at similar datasets, often 'geometry' implies calculating the positive area of the shape and then applying the sign. If a student writes `(-3-1)/2 * 1`, they are using the algebraic property of the integral, not strictly a 'geometric formula' which outputs a positive magnitude. This is a subtle distinction. If I must fail, it's 'misleading'. If I can pass, I will. Let's check if the shape is actually a trapezoid. Yes. Let's check the bounds. 0 to 1. Yes. I will pass.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer returned a non-objectgpt-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.