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Area between curves

Problem 5.147 · medium

Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} - 8 x + 6 \) and \( \displaystyle y = - 2 x - 3 \).

\[ \left(3 - 3 x\right) \left(x + 3\right) = - 3 x^{2} - 6 x + 9 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = -3 and x = 1; between them y = -3*x**2 - 8*x + 6 is on top.
\[ \int\limits_{-3}^{1} \left(- 3 x^{2} - 6 x + 9\right)\, dx = 32 \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( 32 \)

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.

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 intersection points are incorrect; solving -3x^2 - 8x + 6 = -2x - 3 yields x = -1 and x = 3, not -3 and 1. Consequently, the integrand and bounds are wrong, although the final numerical answer coincidentally matches the correct area.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The intersection points are incorrect; solving -3x^2 - 8x + 6 = -2x - 3 yields x = -1 and x = 3, not -3 and 1. Consequently, the integrand and bounds are wrong, although the final numerical answer coincidentally matches the correct area.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The setup in line 1 is incorrect: the difference between the curves is -3x^2 - 6x + 9, but the problem states the curves are y = -3x^2 - 8x + 6 and y = -2x - 3, whose difference is -3x^2 - 6x + 9 only if the linear term was -6x, not -8x. Wait, let's re-calculate: (-3x^2 - 8x + 6) - (-2x - 3) = -3x^2 - 6x + 9. The algebra in the equation line is actually correct for the difference. However, line 1 says '-3*(x - 1)*(x + 3)' which expands to -3(x^2 + 2x - 3) = -3x^2 - 6x + 9. This matches the difference. But the sentence in line 2 claims the curves cross at x=-3 and x=1. Let's check the roots of -3x^2 - 6x + 9 = 0. Dividing by -3: x^2 + 2x - 3 = 0 => (x+3)(x-1)=0. So x=-3 and x=1 are correct. The integral calculation is also correct. Why did I think it was wrong? Let me re-read carefully. Problem: y = -3x^2 - 8x + 6 and y = -2x - 3. Difference: (-3x^2 - 8x + 6) - (-2x - 3) = -3x^2 - 6x + 9. Roots of -3x^2 - 6x + 9 = 0 are x = 1 and x = -3. Integral from -3 to 1 of (-3x^2 - 6x + 9) dx. Antiderivative: -x^3 - 3x^2 + 9x. Evaluate at 1: -1 - 3 + 9 = 5. Evaluate at -3: -(-27) - 3(9) + 9(-3) = 27 - 27 - 27 = -27. Area = 5 - (-27) = 32. The solution is actually correct. The 'unchecked' tag on the equation lines means I shouldn't re-check the algebra, but I should check if the setup models the problem. Line 1 sets up the integrand correctly. Line 2 identifies bounds and top/bottom correctly. Line 3 computes the integral. Wait, look at line 1 again: `-3*(x - 1)*(x + 3), -3*x**2 - 6*x + 9`. This is the integrand. The comment says 'Set the curves equal: top - bottom, factored'. This is slightly confusing phrasing ('Set the curves equal' usually implies finding roots, but here it shows the difference). However, the math is sound. Is there any error? Line 2: 'They cross at x = -3 and x = 1'. Correct. 'between them y = -3*x**2 - 8*x + 6 is on top'. Let's check a point, e.g., x=0. y1(0) = 6. y2(0) = -3. 6 > -3. Correct. The solution seems perfectly correct. Why would it fail? Maybe the prompt implies I should find something wrong? Let's look closer at Line 1. It lists two expressions. The CAS checked them? No, it says 'unchecked'. But the instructions say 'Each line... is either an EQUATION... (its result is shown; do not re-check the algebra of a line marked proved)'. Line 1 is marked 'unchecked'. This usually means the CAS didn't verify it, or it's just a statement. But the instruction says 'do not re-check the algebra of a line marked proved'. Line 1 is NOT marked proved. It is marked 'unchecked'. Does this mean I SHOULD check it? The prompt says: 'Each line of the solution is either an EQUATION, which a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)...'. Line 1 is `[equation, unchecked]`. This likely means it's an equation line but the CAS didn't mark it as 'proved' or 'checked'. Or perhaps 'unchecked' is a typo for 'checked'? If 'unchecked' means I need to check it: -3*(x-1)*(x+3) = -3(x^2 + 2x - 3) = -3x^2 - 6x + 9. The difference of the curves is (-3x^2 - 8x + 6) - (-2x - 3) = -3x^2 - 6x + 9. So the algebra is correct. Is there a subtle error? 'y = - 3 x^{2} - 8 x + 6'. 'y = - 2 x - 3'. Difference: -3x^2 - 6x + 9. Roots: 1, -3. Integral: 32. Everything is correct. I will pass it.
gpt-oss:20b: pass 2026-09-28

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