Home›Calculus 1›Area between curves›Problem 5.171

Area between curves

Problem 5.171 · medium Video

Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 2 x + 16 \) and \( \displaystyle y = - x - 2 \).

\[ \left(9 - 3 x\right) \left(x + 2\right) = - 3 x^{2} + 3 x + 18 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = -2 and x = 3; between them y = -3*x**2 + 2*x + 16 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 \( \frac{125}{2} \)

The curves y = -3x² + 2x + 16 and y = -x - 2 between x = -2 and x = 3. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 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. 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 + 3x + 18, but the problem states the parabola is -3x^2 + 2x + 16. The linear term in the difference should be 3x (from 2x - (-x)), but the constant term in the difference is 18 (from 16 - (-2)), which matches. However, the factored form in line 1 claims -3*(x-3)*(x+2) equals -3x^2 + 3x + 18. Let's check: -3(x^2 - x - 6) = -3x^2 + 3x + 18. This algebra is correct. But wait, does the parabola y = -3x^2 + 2x + 16 intersect y = -x - 2 at x = -2 and x = 3? Let's check: -3(-2)^2 + 2(-2) + 16 = -12 - 4 + 16 = 0. Line: -(-2) - 2 = 0. Match. At x=3: -3(9) + 6 + 16 = -27 + 22 = -5. Line: -3 - 2 = -5. Match. So the intersection points are correct. The difference function is (-3x^2 + 2x + 16) - (-x - 2) = -3x^2 + 3x + 18. This matches line 1. So the setup is actually correct. Why did I think it was wrong? I misread the problem statement's parabola coefficient or something? No, the problem says y = -3x^2 + 2x + 16. The difference is indeed -3x^2 + 3x + 18. The roots of -3x^2 + 3x + 18 = 0 are x = 3 and x = -2. The integral is correct. The answer 125/2 is correct. Wait, let me re-read the prompt's instruction. 'Find what is wrong'. Is there a subtle error? Line 1 says 'Set the curves equal: top - bottom, factored.' The equation shown is '-3*(x - 3)*(x + 2) = -3*x**2 + 3*x + 18'. This is an identity, not the setup of the integral. But it is marked [equation, proved]. Line 2 states the bounds and which is on top. Line 3 computes the integral. Everything seems correct. Let me double check the arithmetic. Integral of -3x^2 + 3x + 18 from -2 to 3. Antiderivative: -x^3 + 1.5x^2 + 18x. Eval at 3: -27 + 1.5(9) + 54 = -27 + 13.5 + 54 = 40.5. Eval at -2: -(-8) + 1.5(4) - 36 = 8 + 6 - 36 = -22. Difference: 40.5 - (-22) = 62.5 = 125/2. The solution is correct. Why would it fail? Maybe the sentence 'Set the curves equal' is misleading because the equation shown is an expansion, not the equality of the two original functions? Or maybe the factored form is presented as the 'top - bottom' expression, which is fine. Is there a style issue? 'Set the curves equal' usually implies finding intersections, but the equation shown is the integrand. This is slightly confusing phrasing but not an error. Let's look closer. Is the parabola actually on top? Vertex of parabola is at x = -b/2a = -2/-6 = 1/3. Value at 1/3: -3(1/9) + 2/3 + 16 = -1/3 + 2/3 + 16 = 16.33. Line at 1/3: -1/3 - 2 = -2.33. Parabola is on top. Correct. I see no error. I will mark it pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The setup in line 1 is incorrect: the difference between the curves is -3x^2 + 3x + 18, but the problem states the parabola is -3x^2 + 2x + 16. The linear term in the difference should be 3x (from 2x - (-x)), but the constant term in the difference is 18 (from 16 - (-2)), which matches. However, the factored form in line 1 claims -3*(x-3)*(x+2) equals -3x^2 + 3x + 18. Let's check: -3(x^2 - x - 6) = -3x^2 + 3x + 18. This algebra is correct. But wait, does the parabola y = -3x^2 + 2x + 16 intersect y = -x - 2 at x = -2 and x = 3? Let's check: -3(-2)^2 + 2(-2) + 16 = -12 - 4 + 16 = 0. Line: -(-2) - 2 = 0. Match. At x=3: -3(9) + 6 + 16 = -27 + 22 = -5. Line: -3 - 2 = -5. Match. So the intersection points are correct. The difference function is (-3x^2 + 2x + 16) - (-x - 2) = -3x^2 + 3x + 18. This matches line 1. So the setup is actually correct. Why did I think it was wrong? I misread the problem statement's parabola coefficient or something? No, the problem says y = -3x^2 + 2x + 16. The difference is indeed -3x^2 + 3x + 18. The roots of -3x^2 + 3x + 18 = 0 are x = 3 and x = -2. The integral is correct. The answer 125/2 is correct. Wait, let me re-read the prompt's instruction. 'Find what is wrong'. Is there a subtle error? Line 1 says 'Set the curves equal: top - bottom, factored.' The equation shown is '-3*(x - 3)*(x + 2) = -3*x**2 + 3*x + 18'. This is an identity, not the setup of the integral. But it is marked [equation, proved]. Line 2 states the bounds and which is on top. Line 3 computes the integral. Everything seems correct. Let me double check the arithmetic. Integral of -3x^2 + 3x + 18 from -2 to 3. Antiderivative: -x^3 + 1.5x^2 + 18x. Eval at 3: -27 + 1.5(9) + 54 = -27 + 13.5 + 54 = 40.5. Eval at -2: -(-8) + 1.5(4) - 36 = 8 + 6 - 36 = -22. Difference: 40.5 - (-22) = 62.5 = 125/2. The solution is correct. Why would it fail? Maybe the sentence 'Set the curves equal' is misleading because the equation shown is an expansion, not the equality of the two original functions? Or maybe the factored form is presented as the 'top - bottom' expression, which is fine. Is there a style issue? 'Set the curves equal' usually implies finding intersections, but the equation shown is the integrand. This is slightly confusing phrasing but not an error. Let's look closer. Is the parabola actually on top? Vertex of parabola is at x = -b/2a = -2/-6 = 1/3. Value at 1/3: -3(1/9) + 2/3 + 16 = -1/3 + 2/3 + 16 = 16.33. Line at 1/3: -1/3 - 2 = -2.33. Parabola is on top. Correct. I see no error. I will mark it pass.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The intersection points are incorrect; solving -3x^2 + 2x + 16 = -x - 2 yields x = -2 and x = 8, not x = 3. Consequently, the integration bounds are wrong.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The intersection points are incorrect; solving -3x^2 + 2x + 16 = -x - 2 yields x = 2 and x = -2, not x = 3 and x = -2. Consequently, the integration bounds and the resulting area are wrong.
gpt-oss:20b: pass 2026-09-29

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 2026-10-04 with SymPy 1.14.0.