Definite integrals from area formulas
Problem 4.1368 · easy Video
Evaluate \( \displaystyle \int_{4}^{5} 3 - x \, dx \) using a formula from geometry.
- The region between the graph and the x-axis is a trapezoid, below the axis.Reviewed
- \[ \frac{\left(-4 + 5\right) \left(-2 - 1\right)}{2} = - \frac{3}{2} \]Its (signed) area.✓ Proved
- \[ \int\limits_{4}^{5} \left(3 - x\right)\, dx = - \frac{3}{2} \]Integrating directly gives the same.✓ Proved
Answer \( - \frac{3}{2} \)
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 geometric shape as a trapezoid and applies the area formula with appropriate signs to account for the region being below the x-axis. The logic is sound.
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 appropriate signs to account for the region being below the x-axis. The logic is sound.gpt-oss:20b: pass 2026-10-29qwen3.6:27b-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 for the definite integral.gpt-oss:20b: pass 2026-10-28qwen3.6:27b-mlx: fail (error) 2026-10-28 — The solution incorrectly identifies the region as a trapezoid; the graph of y = 3 - x is a line, so the region between the graph and the x-axis from x=4 to x=5 is a triangle (since the function does not cross the x-axis in this interval, but the 'height' at x=3 is 0, making the shape from x=3 to x=5 a triangle, or simply calculating the area of the trapezoid formed by the vertical lines x=4, x=5, the x-axis, and the line y=3-x is valid, but the description 'below the axis' is correct, however, the geometric shape bounded by x=4, x=5, y=0 and y=3-x is indeed a trapezoid with parallel vertical sides? No, the parallel sides are the vertical segments at x=4 and x=5? No, the parallel sides of a trapezoid in this context are usually the vertical drops to the x-axis if we view it as a standard area under curve, but strictly speaking, the region is a trapezoid with parallel sides being the vertical lines x=4 and x=5? No, the parallel sides are the vertical segments f(4) and f(5) if we rotate it, but standardly, the area under a line is a trapezoid with parallel sides perpendicular to the x-axis? No, the parallel sides are the vertical segments at the endpoints. Wait, a trapezoid has one pair of parallel sides. The vertical segments at x=4 and x=5 are parallel. So it is a trapezoid. However, the calculation in line 2 uses heights -2 and -1. The width is 1. Area = 1/2 * (|-2| + |-1|) * 1 = 1.5. Since it is below the axis, the signed area is -1.5. The calculation (-4+5)*(-2-1)/2 = 1 * (-3)/2 = -1.5. This is correct. Why is it failing? Let's re-read carefully. 'The region between the graph and the x-axis is a trapezoid'. The vertices are (4,0), (5,0), (5,-2), (4,-1). This is a trapezoid with parallel vertical sides? No, the sides x=4 and x=5 are vertical and parallel. The other two sides are the x-axis (horizontal) and the line y=3-x (slanted). This is a right trapezoid. The formula for the area of a trapezoid is 1/2 * (sum of parallel sides) * height. Here the parallel sides are the vertical segments of length 1 and 2. The 'height' of the trapezoid (distance between parallel sides) is 5-4=1. So Area = 1/2 * (1+2) * 1 = 1.5. Signed area is -1.5. The solution says 'trapezoid, below the axis'. This is geometrically correct. Is there a subtle error? Maybe the prompt implies using a simpler geometric shape? No, trapezoid is fine. Let's look at the equation line 2: `(-4 + 5)*(-2 - 1)/2`. This calculates `width * (sum of heights) / 2`. This is the correct formula for the area of a trapezoid with vertical parallel sides. The result is -3/2. The integral is -3/2. Everything seems correct. Why would it fail? Perhaps the 'sentence' is misleading because it's technically a trapezoid, but often students are taught to split it into a rectangle and a triangle? No, trapezoid is standard. Let's look closer at the prompt's constraint: 'using a formula from geometry'. The solution does this. Is there an error in the sign? The problem asks to evaluate the integral. The integral is negative. The geometric area is positive, but the signed area is negative. The solution correctly assigns the negative sign. Wait, look at line 1: 'The region ... is a trapezoid, below the axis.' This is a true statement. Line 2 calculates the signed area. Line 3 confirms. It seems correct. Is there a 'style' issue? Or is it actually correct? Let me double check the 'trapezoid' definition. A trapezoid is a quadrilateral with at least one pair of parallel sides. The region bounded by x=4, x=5, y=0, y=3-x has vertices (4,0), (5,0), (5,-2), (4,-1). The side from (4,0) to (4,-1) is vertical. The side from (5,0) to (5,-2) is vertical. They are parallel. So it is a trapezoid. The calculation is correct. The verdict should be pass? Let me look for a hidden trap. 'Evaluate ... using a formula from geometry'. Maybe the trap is that the region is not a simple geometric shape? No, it is. Maybe the trap is that the student should have used a triangle? If we extend the line to x=3, y=0, the region from 3 to 5 is a triangle. But the integral is from 4 to 5. So it's a trapezoid. Or a large triangle minus a small triangle. Using the trapezoid formula directly is valid. I will mark it as pass unless I find a specific error. Wait, looking at similar problems, sometimes 'formula from geometry' implies using basic shapes like rectangles and triangles, and 'trapezoid' might be considered a derived formula? No, the area of a trapezoid is a standard geometric formula. Let's look at the equation syntax. `(-4 + 5)*(-2 - 1)/2`. This is `1 * -3 / 2 = -1.5`. Correct. I see no error. I will output pass.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.