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Area between curves

Problem 5.100 · medium

Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 2 x + 21 \) and \( \displaystyle y = 3 - x \).

\[ \left(9 - 3 x\right) \left(x + 2\right) = - 3 x^{2} + 3 x + 18 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = -2 and x = 3; between them y = -3*x**2 + 2*x + 21 is on top.
\[ \int\limits_{-2}^{3} \left(- 3 x^{2} + 3 x + 18\right)\, dx = \frac{125}{2} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{125}{2} \)

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 setup in line 1 is incorrect: the difference (top - bottom) is -3x^2 + 3x + 18, but the problem states the parabola is y = -3x^2 + 2x + 21. The correct difference is -3x^2 + 3x + 18 only if the parabola were -3x^2 + 2x + 21 and the line 3-x, wait: (-3x^2 + 2x + 21) - (3 - x) = -3x^2 + 3x + 18. The algebra in line 1 is actually correct for the given functions. However, line 1 claims the factored form is -3*(x-3)*(x+2). Let's check: -3(x^2 - x - 6) = -3x^2 + 3x + 18. This matches. The roots are indeed -2 and 3. The integral calculation is correct. Wait, let me re-read the prompt's solution line 1 carefully. It says '-3*(x - 3)*(x + 2), -3*x**2 + 3*x + 18'. This is the integrand. The sentence says 'Set the curves equal: top - bottom, factored.' This is slightly confusing phrasing but the math holds. Is there an error? Let's check the area calculation. Integral of -3x^2 + 3x + 18 from -2 to 3. Antiderivative: -x^3 + 1.5x^2 + 18x. Eval at 3: -27 + 13.5 + 54 = 40.5. Eval at -2: 8 + 6 - 36 = -22. 40.5 - (-22) = 62.5 = 125/2. The math is correct. Why would it fail? Ah, look at line 1 again. The text says 'Set the curves equal: top - bottom'. Setting curves equal finds intersections. Top - bottom finds the integrand. These are two different things. The sentence conflates finding roots with setting up the integrand. But is it an error? It's a bit sloppy. Let's look closer. Is the parabola actually on top? Vertex at x = -2/(-6) = 1/3. At x=0, y=21. Line y=3. Yes, parabola is on top. The solution seems mathematically sound. Let me check the 'proved' tag. The equation line is marked proved. The sentence is what I review. Sentence 2: 'They cross at x = -2 and x = 3; between them y = ... is on top.' This is correct. Sentence 3 is an equation. So where is the error? Maybe the phrasing in line 1's comment? No, I only review sentences. Sentence 2 is the only sentence. It is correct. Wait, did I miss something? 'Set the curves equal: top - bottom, factored.' This is part of the equation line's comment, not a sentence. The sentences are explicitly marked [sentence]. There is only one sentence: line 2. It is correct. So why would it be a fail? Let me re-read the instructions. 'Each line ... is either an EQUATION ... or a SENTENCE'. Line 1 is an equation. Line 2 is a sentence. Line 3 is an equation. I only review sentences. Sentence 2 is correct. Verdict: pass? Let me double check the problem statement vs solution. Problem: y = -3x^2 + 2x + 21. Solution integrand: -3x^2 + 3x + 18. Difference: (-3x^2 + 2x + 21) - (3 - x) = -3x^2 + 3x + 18. Correct. Roots of -3x^2 + 3x + 18 = 0 => -3(x^2 - x - 6) = 0 => (x-3)(x+2)=0. Correct. Integral value 125/2. Correct. The sentence is correct. Pass.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-26 — The setup in line 1 is incorrect: the difference (top - bottom) is -3x^2 + 3x + 18, but the problem states the parabola is y = -3x^2 + 2x + 21. The correct difference is -3x^2 + 3x + 18 only if the parabola were -3x^2 + 2x + 21 and the line 3-x, wait: (-3x^2 + 2x + 21) - (3 - x) = -3x^2 + 3x + 18. The algebra in line 1 is actually correct for the given functions. However, line 1 claims the factored form is -3*(x-3)*(x+2). Let's check: -3(x^2 - x - 6) = -3x^2 + 3x + 18. This matches. The roots are indeed -2 and 3. The integral calculation is correct. Wait, let me re-read the prompt's solution line 1 carefully. It says '-3*(x - 3)*(x + 2), -3*x**2 + 3*x + 18'. This is the integrand. The sentence says 'Set the curves equal: top - bottom, factored.' This is slightly confusing phrasing but the math holds. Is there an error? Let's check the area calculation. Integral of -3x^2 + 3x + 18 from -2 to 3. Antiderivative: -x^3 + 1.5x^2 + 18x. Eval at 3: -27 + 13.5 + 54 = 40.5. Eval at -2: 8 + 6 - 36 = -22. 40.5 - (-22) = 62.5 = 125/2. The math is correct. Why would it fail? Ah, look at line 1 again. The text says 'Set the curves equal: top - bottom'. Setting curves equal finds intersections. Top - bottom finds the integrand. These are two different things. The sentence conflates finding roots with setting up the integrand. But is it an error? It's a bit sloppy. Let's look closer. Is the parabola actually on top? Vertex at x = -2/(-6) = 1/3. At x=0, y=21. Line y=3. Yes, parabola is on top. The solution seems mathematically sound. Let me check the 'proved' tag. The equation line is marked proved. The sentence is what I review. Sentence 2: 'They cross at x = -2 and x = 3; between them y = ... is on top.' This is correct. Sentence 3 is an equation. So where is the error? Maybe the phrasing in line 1's comment? No, I only review sentences. Sentence 2 is the only sentence. It is correct. Wait, did I miss something? 'Set the curves equal: top - bottom, factored.' This is part of the equation line's comment, not a sentence. The sentences are explicitly marked [sentence]. There is only one sentence: line 2. It is correct. So why would it be a fail? Let me re-read the instructions. 'Each line ... is either an EQUATION ... or a SENTENCE'. Line 1 is an equation. Line 2 is a sentence. Line 3 is an equation. I only review sentences. Sentence 2 is correct. Verdict: pass? Let me double check the problem statement vs solution. Problem: y = -3x^2 + 2x + 21. Solution integrand: -3x^2 + 3x + 18. Difference: (-3x^2 + 2x + 21) - (3 - x) = -3x^2 + 3x + 18. Correct. Roots of -3x^2 + 3x + 18 = 0 => -3(x^2 - x - 6) = 0 => (x-3)(x+2)=0. Correct. Integral value 125/2. Correct. The sentence is correct. Pass.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The setup in line 1 is incorrect: the difference between the curves is -3x^2 + 3x + 18, but the problem states the quadratic is -3x^2 + 2x + 21. The correct difference is -3x^2 + 3x + 18, which implies the quadratic should have been -3x^2 + 2x + 21? Wait. (-3x^2 + 2x + 21) - (3 - x) = -3x^2 + 3x + 18. The algebra in line 1 is actually correct for the given functions. However, line 1 says '-3*(x - 3)*(x + 2), -3*x**2 + 3*x + 18'. Let's check the roots of -3x^2 + 3x + 18. Roots are x = [-3 +/- sqrt(9 - 4(-3)(18))]/-6 = [-3 +/- sqrt(9 + 216)]/-6 = [-3 +/- 15]/-6. x1 = 12/-6 = -2. x2 = -18/-6 = 3. So the roots are correct. The integration result 125/2 is correct for the integral of -3x^2 + 3x + 18 from -2 to 3. Wait, let me re-read the prompt's solution carefully. Line 1 says: '-3*(x - 3)*(x + 2), -3*x**2 + 3*x + 18'. This is the integrand. The problem asks for area between y = -3x^2 + 2x + 21 and y = 3 - x. Top - Bottom = (-3x^2 + 2x + 21) - (3 - x) = -3x^2 + 3x + 18. This matches. The roots are -2 and 3. The integral is 125/2. Why did I think it was wrong? Let me check the stated answer. 125/2. Let me check the integral value. Int(-3x^2 + 3x + 18) = -x^3 + 1.5x^2 + 18x. Evaluated at 3: -27 + 13.5 + 54 = 40.5. Evaluated at -2: 8 + 6 - 36 = -22. Difference: 40.5 - (-22) = 62.5 = 125/2. The solution is actually correct. Is there a subtle error? 'Set the curves equal: top - bottom, factored.' The sentence in line 2 says 'They cross at x = -2 and x = 3; between them y = -3*x**2 + 2*x + 21 is on top.' This is true. The solution seems perfectly sound. Why would it fail? Maybe the prompt implies I should find an error? Or maybe I missed something. Let's look at line 1 again. '-3*(x - 3)*(x + 2)'. This expands to -3(x^2 - x - 6) = -3x^2 + 3x + 18. Correct. The problem statement: y = -3x^2 + 2x + 21. y = 3 - x. Difference: -3x^2 + 3x + 18. Correct. Roots: -2, 3. Correct. Integral: 125/2. Correct. Verdict: pass.
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 2026-09-26 with SymPy 1.14.0.