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Area between curves

Problem 5.4 · medium Video

Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 10 x - 1 \) and \( \displaystyle y = - 2 x - 1 \).

\[ - 3 x \left(x - 4\right) = - 3 x^{2} + 12 x \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = 0 and x = 4; between them y = -3*x**2 + 10*x - 1 is on top.
\[ \int\limits_{0}^{4} \left(- 3 x^{2} + 12 x\right)\, dx = 32 \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( 32 \)

The curves y = -3x² + 10x - 1 and y = -2x - 1 between x = 0 and x = 4. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 32.

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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Not checked	—	a sentence; read, not computed
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 of \|top − bottom\| gives the same area

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The setup in line 1 is incorrect: the difference between the curves is (-3x^2 + 10x - 1) - (-2x - 1) = -3x^2 + 12x, but the text claims this is 'top - bottom, factored' while showing -3*x*(x-4). While the factoring is correct for the difference, the sentence in line 2 claims the curves cross at x=0 and x=4. Let's check: -3x^2 + 10x - 1 = -2x - 1 => -3x^2 + 12x = 0 => -3x(x-4)=0. So x=0 and x=4 are correct intersection points. The integral calculation is correct. However, line 1 says '-3*x*(x - 4), -3*x**2 + 12*x -- Set the curves equal: top − bottom, factored.' This is slightly confusingly phrased but mathematically the difference is correct. Wait, let me re-read carefully. The problem asks for the area. The difference is indeed -3x^2 + 12x. The integral of this from 0 to 4 is 32. The solution is actually correct. Why did I think it was wrong? Let me double check the 'top' claim. At x=2, y1 = -3(4) + 20 - 1 = 11. y2 = -4 - 1 = -5. 11 > -5. So y1 is on top. The difference is y1 - y2. The integral is correct. The verdict should be pass? No, wait. Line 1 says 'Set the curves equal: top - bottom'. Setting curves equal gives the intersection points, not the integrand. The integrand is the difference. The comment is conflating finding intersections with setting up the integrand. But the math shown `-3*x**2 + 12*x` is the correct integrand. Is it misleading? It says 'Set the curves equal' which usually implies y1=y2. But then it shows the difference. This is a 'style' or 'misleading' issue in the comment, but the math is right. Let's look closer. 'Set the curves equal: top - bottom'. This is a category error in the description. You don't set them equal to get top-bottom. You subtract them. However, the result is correct. Is it 'misleading'? A student might think 'setting equal' produces the difference. That is false. So it is misleading. But is it an error in the solution? The solution steps are: 1. Compute difference. 2. Find bounds. 3. Integrate. The text in 1 is poorly phrased. Let's look at the options. 'error' means a step is wrong. The step (the equation) is proved correct. The sentence is the comment. The comment is misleading. I will mark it as misleading.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The setup in line 1 is incorrect: the difference between the curves is (-3x^2 + 10x - 1) - (-2x - 1) = -3x^2 + 12x, but the text claims this is 'top - bottom, factored' while showing -3*x*(x-4). While the factoring is correct for the difference, the sentence in line 2 claims the curves cross at x=0 and x=4. Let's check: -3x^2 + 10x - 1 = -2x - 1 => -3x^2 + 12x = 0 => -3x(x-4)=0. So x=0 and x=4 are correct intersection points. The integral calculation is correct. However, line 1 says '-3*x*(x - 4), -3*x**2 + 12*x -- Set the curves equal: top − bottom, factored.' This is slightly confusingly phrased but mathematically the difference is correct. Wait, let me re-read carefully. The problem asks for the area. The difference is indeed -3x^2 + 12x. The integral of this from 0 to 4 is 32. The solution is actually correct. Why did I think it was wrong? Let me double check the 'top' claim. At x=2, y1 = -3(4) + 20 - 1 = 11. y2 = -4 - 1 = -5. 11 > -5. So y1 is on top. The difference is y1 - y2. The integral is correct. The verdict should be pass? No, wait. Line 1 says 'Set the curves equal: top - bottom'. Setting curves equal gives the intersection points, not the integrand. The integrand is the difference. The comment is conflating finding intersections with setting up the integrand. But the math shown `-3*x**2 + 12*x` is the correct integrand. Is it misleading? It says 'Set the curves equal' which usually implies y1=y2. But then it shows the difference. This is a 'style' or 'misleading' issue in the comment, but the math is right. Let's look closer. 'Set the curves equal: top - bottom'. This is a category error in the description. You don't set them equal to get top-bottom. You subtract them. However, the result is correct. Is it 'misleading'? A student might think 'setting equal' produces the difference. That is false. So it is misleading. But is it an error in the solution? The solution steps are: 1. Compute difference. 2. Find bounds. 3. Integrate. The text in 1 is poorly phrased. Let's look at the options. 'error' means a step is wrong. The step (the equation) is proved correct. The sentence is the comment. The comment is misleading. I will mark it as misleading.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The intersection points are incorrect. Solving -3x^2 + 10x - 1 = -2x - 1 yields -3x^2 + 12x = 0, so x = 0 and x = 4. Wait, let me re-check. -3x^2 + 10x - 1 = -2x - 1 => -3x^2 + 12x = 0 => -3x(x - 4) = 0. So x=0 and x=4 are correct. The integral of -3x^2 + 12x from 0 to 4 is [-x^3 + 6x^2] from 0 to 4 = (-64 + 96) - 0 = 32. The solution is actually correct. Why did I think it was wrong? Let me re-read carefully. Problem: y = -3x^2 + 10x - 1 and y = -2x - 1. Difference: (-3x^2 + 10x - 1) - (-2x - 1) = -3x^2 + 12x. Roots: -3x(x-4)=0 -> x=0, 4. Integral: int_0^4 (-3x^2 + 12x) dx = [-x^3 + 6x^2]_0^4 = -64 + 96 = 32. The solution is correct. I will mark it as pass.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The setup in line 1 is incorrect: the difference between the curves is (-3x^2 + 10x - 1) - (-2x - 1) = -3x^2 + 12x, but the solution text claims this is 'top - bottom' while the equation shown is just the simplified difference. More critically, line 2 states the curves cross at x=0 and x=4, which is correct for -3x^2+12x=0, but the reasoning implies the integrand is -3x^2+12x. However, the text in line 1 says '-3*x*(x - 4), -3*x**2 + 12*x' which is the factorization of the difference. The main error is in the description of line 1: it says 'Set the curves equal: top - bottom'. Setting curves equal finds intersections. The difference is top - bottom. The text conflates these steps. But wait, looking closer at line 1: '-3*x*(x - 4)' is indeed -3x^2 + 12x. The difference is correct. The bounds are correct. The integral is correct. Is there an error? Let's re-read carefully. Curve 1: y = -3x^2 + 10x - 1 Curve 2: y = -2x - 1 Difference: (-3x^2 + 10x - 1) - (-2x - 1) = -3x^2 + 12x. Roots of difference: -3x(x-4) = 0 => x=0, x=4. Between 0 and 4, test x=2: -3(4) + 24 = 12 > 0. So Curve 1 is on top. Integral of (-3x^2 + 12x) from 0 to 4 is [-x^3 + 6x^2] from 0 to 4 = -64 + 96 = 32. The solution is mathematically correct. Why did I think it was wrong? Line 1 text: 'Set the curves equal: top - bottom, factored.' This is a confusing sentence. You don't 'set the curves equal' to get 'top - bottom'. You subtract them. Setting them equal gives the intersection points. The text is garbled but the math is right. However, the prompt asks to find what is wrong. Is 'Set the curves equal' a valid description for finding the integrand? No. It's a description for finding bounds. The line computes the difference. The text is misleading/wrong in its description of the operation. Let's look at line 2: 'They cross at x = 0 and x = 4'. This follows from setting the difference to 0. The error is in the text of line 1. It says 'Set the curves equal' but performs subtraction. This is a 'misleading' or 'error' in the explanation. Given the strict adversarial role, claiming 'Set the curves equal' results in the expression for the area integrand is false. Setting them equal results in an equation -3x^2 + 12x = 0. Verdict: fail. Severity: misleading (or error in reasoning description). A student might think 'setting curves equal' gives the integrand.
gpt-oss:20b: pass 2026-09-26

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/area_between_curves, checked 2026-09-28 with SymPy 1.14.0.