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Definite integrals from area formulas

Problem 4.1273 · easy Video

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

The region between the graph and the x-axis is two triangles meeting at x = 0.
Reviewed
\[ \frac{1^{2}}{2} + \frac{3^{2}}{2} = 5 \]
Its (signed) area.✓ Proved
\[ \int\limits_{-1}^{3} \left|{x}\right|\, dx = 5 \]
Integrating directly gives the same.✓ Proved

Answer \( 5 \)

The graph of f(x) = Abs(x) from x = -1 to x = 3. Rectangles under the curve get thinner until they fill the area, 5.

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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the geometric shapes (two right triangles) and calculates their areas accurately to evaluate the definite integral of |x|.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the geometric shapes (two right triangles) and calculates their areas accurately to evaluate the definite integral of |x|.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the geometric shapes (two triangles) and calculates their areas to find the integral of the absolute value function. The reasoning is sound and the result is correct.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (misleading) 2026-10-17 — The solution claims the area is the sum of two triangles, but the triangle on the left has base 1 and height 1, so its area is 1/2, not 1/2 * 1^2 (which is correct numerically but the formula used in line 2 implies a generic x^2/2 which is confusing). More importantly, line 2 calculates 1^2/2 + 3^2/2 = 0.5 + 4.5 = 5. This is numerically correct. However, the sentence in step 1 says 'two triangles meeting at x=0'. The triangle on the left is from -1 to 0 (base 1, height 1, area 0.5). The triangle on the right is from 0 to 3 (base 3, height 3, area 4.5). The calculation in line 2 is correct. Wait, let me re-read carefully. 1^2/2 is 0.5. 3^2/2 is 4.5. Sum is 5. The logic is sound. Why would it fail? Ah, the prompt says 'using a formula from geometry'. The solution does use geometry. Is there a subtle error? The problem asks to evaluate the integral. The solution provides the answer 5. The steps are: 1. Identify regions. 2. Calculate areas. 3. Verify. This seems correct. Let me look for 'misleading' or 'style'. The sentence 'The region... is two triangles' is slightly imprecise because the region is the union of two triangles, but that's standard phrasing. Is there an error in the bounds? Integral from -1 to 3. Left part: -1 to 0. Right part: 0 to 3. Correct. Heights: |-1|=1, |3|=3. Correct. Areas: 0.5*1*1 = 0.5. 0.5*3*3 = 4.5. Sum = 5. Correct. Wait, look at line 2: `1**2/2 + 3**2/2 = 5`. This is `0.5 + 4.5 = 5`. Correct. Is it possible the reviewer wants me to find that the solution doesn't explicitly state the geometric formula A = 1/2 * base * height? It just writes the numbers. That might be 'style' or 'misleading' if a student doesn't see the connection. But the sentence says 'two triangles'. Let's look closer at the prompt's instruction: 'Check that... each conclusion follows from the lines before it'. Line 2 is an equation. Line 1 is a sentence. Does Line 2 follow from Line 1? Line 1 identifies the shapes. Line 2 calculates the area. It skips the explicit statement of the triangle area formula, but it's a standard calculation. However, often in these adversarial reviews, 'misleading' is used if the explanation is insufficient to teach the method. The solution jumps from 'it's two triangles' to '1^2/2 + 3^2/2'. A student might not know where the squares came from. It's better to say 'The area of a triangle is 1/2 * base * height. For the left triangle, base=1, height=1...'. Without this, the jump is large. Also, is there a sign error? No, absolute value makes areas positive. Let's consider if '1**2/2' is interpreted as (1^2)/2 or 1^(2/2)? Standard order of operations is (1^2)/2. Actually, there is a potential issue with the phrase 'Its (signed) area' in line 2. The integral of |x| is always positive, so 'signed area' is a confusing term here, as there are no negative contributions. The area is just 'area'. Calling it 'signed area' might mislead a student into thinking about cancellation, which doesn't happen with |x|. This is 'misleading'. Another possibility: The problem asks to use a formula from geometry. The solution does so. Let's stick with 'misleading' due to the term 'signed area' for a strictly positive integrand, and the lack of explicit geometric formula application.
gpt-oss:20b: pass 2026-10-17

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 2026-10-19 with SymPy 1.14.0.