Home›Calculus 1›Area between curves›Problem 5.65

Area between curves

Problem 5.65 · medium Video

Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 11 x - 9 \) and \( \displaystyle y = x + 3 \).

\[ \left(6 - 2 x\right) \left(x - 2\right) = - 2 x^{2} + 10 x - 12 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = 2 and x = 3; between them y = -2*x**2 + 11*x - 9 is on top.
\[ \int\limits_{2}^{3} \left(- 2 x^{2} + 10 x - 12\right)\, dx = \frac{1}{3} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{1}{3} \)

The curves y = -2x² + 11x - 9 and y = x + 3 between x = 2 and x = 3. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 1/3.

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 incorrectly computes the difference between the curves as -2x^2 + 10x - 12, whereas the correct difference (top minus bottom) is -2x^2 + 10x - 12. Wait, let me re-calculate. Top: -2x^2 + 11x - 9. Bottom: x + 3. Difference: (-2x^2 + 11x - 9) - (x + 3) = -2x^2 + 10x - 12. This part is actually correct algebraically. Let's check the roots. -2x^2 + 10x - 12 = 0 => -2(x^2 - 5x + 6) = 0 => -2(x-2)(x-3) = 0. Roots are 2 and 3. Correct. Let's check the integral. Integral of -2x^2 + 10x - 12 from 2 to 3. Antiderivative: -2/3 x^3 + 5x^2 - 12x. Evaluated at 3: -2/3(27) + 5(9) - 12(3) = -18 + 45 - 36 = -9. Evaluated at 2: -2/3(8) + 5(4) - 12(2) = -16/3 + 20 - 24 = -16/3 - 4 = -28/3. Result: -9 - (-28/3) = -27/3 + 28/3 = 1/3. The calculation is correct. Why did I think it was wrong? Let me re-read the prompt carefully. 'Find the area of the region bounded by y = -2x^2 + 11x - 9 and y = x + 3'. The solution calculates the integral of (top - bottom) which is positive between 2 and 3? Let's check a point between 2 and 3, e.g., x=2.5. Top: -2(6.25) + 11(2.5) - 9 = -12.5 + 27.5 - 9 = 6. Bottom: 2.5 + 3 = 5.5. Top > Bottom. So the integrand is positive. The integral result 1/3 is positive. Everything seems correct. Is there a subtle error? Let's look at line 1 again. '-2*(x - 3)*(x - 2), -2*x**2 + 10*x - 12'. This is the factored form and expanded form of the difference. Line 2 says they cross at 2 and 3 and the parabola is on top. Line 3 integrates the difference. The result is 1/3. The stated answer is 1/3. It seems correct. Wait, did I miss something? Let me double check the arithmetic one more time. Integral from 2 to 3 of (-2x^2 + 10x - 12) dx. Let u = x - 2.5? No, just standard integration. [-2/3 x^3 + 5x^2 - 12x]_2^3. F(3) = -18 + 45 - 36 = -9. F(2) = -16/3 + 20 - 24 = -16/3 - 4 = -28/3. F(3) - F(2) = -27/3 + 28/3 = 1/3. The math is solid. The logic is solid. The setup is solid. Why would this fail? Maybe the prompt implies I should find an error? Or maybe it's a 'pass'? Let me look really closely at the text. 'Set the curves equal: top - bottom, factored.' This is a comment on the equation line. The equation line shows the factored form and the expanded form. It doesn't explicitly state 'Difference = ...'. But line 3 says 'Area = Integral(top - bottom)'. This implies the integrand in line 3 is the top minus bottom. Line 1 provides the expression for top minus bottom. It seems consistent. I will mark it as pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The setup in line 1 incorrectly computes the difference between the curves as -2x^2 + 10x - 12, whereas the correct difference (top minus bottom) is -2x^2 + 10x - 12. Wait, let me re-calculate. Top: -2x^2 + 11x - 9. Bottom: x + 3. Difference: (-2x^2 + 11x - 9) - (x + 3) = -2x^2 + 10x - 12. This part is actually correct algebraically. Let's check the roots. -2x^2 + 10x - 12 = 0 => -2(x^2 - 5x + 6) = 0 => -2(x-2)(x-3) = 0. Roots are 2 and 3. Correct. Let's check the integral. Integral of -2x^2 + 10x - 12 from 2 to 3. Antiderivative: -2/3 x^3 + 5x^2 - 12x. Evaluated at 3: -2/3(27) + 5(9) - 12(3) = -18 + 45 - 36 = -9. Evaluated at 2: -2/3(8) + 5(4) - 12(2) = -16/3 + 20 - 24 = -16/3 - 4 = -28/3. Result: -9 - (-28/3) = -27/3 + 28/3 = 1/3. The calculation is correct. Why did I think it was wrong? Let me re-read the prompt carefully. 'Find the area of the region bounded by y = -2x^2 + 11x - 9 and y = x + 3'. The solution calculates the integral of (top - bottom) which is positive between 2 and 3? Let's check a point between 2 and 3, e.g., x=2.5. Top: -2(6.25) + 11(2.5) - 9 = -12.5 + 27.5 - 9 = 6. Bottom: 2.5 + 3 = 5.5. Top > Bottom. So the integrand is positive. The integral result 1/3 is positive. Everything seems correct. Is there a subtle error? Let's look at line 1 again. '-2*(x - 3)*(x - 2), -2*x**2 + 10*x - 12'. This is the factored form and expanded form of the difference. Line 2 says they cross at 2 and 3 and the parabola is on top. Line 3 integrates the difference. The result is 1/3. The stated answer is 1/3. It seems correct. Wait, did I miss something? Let me double check the arithmetic one more time. Integral from 2 to 3 of (-2x^2 + 10x - 12) dx. Let u = x - 2.5? No, just standard integration. [-2/3 x^3 + 5x^2 - 12x]_2^3. F(3) = -18 + 45 - 36 = -9. F(2) = -16/3 + 20 - 24 = -16/3 - 4 = -28/3. F(3) - F(2) = -27/3 + 28/3 = 1/3. The math is solid. The logic is solid. The setup is solid. Why would this fail? Maybe the prompt implies I should find an error? Or maybe it's a 'pass'? Let me look really closely at the text. 'Set the curves equal: top - bottom, factored.' This is a comment on the equation line. The equation line shows the factored form and the expanded form. It doesn't explicitly state 'Difference = ...'. But line 3 says 'Area = Integral(top - bottom)'. This implies the integrand in line 3 is the top minus bottom. Line 1 provides the expression for top minus bottom. It seems consistent. I will mark it as pass.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: inconclusive 2026-09-27 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The setup in line 1 is incorrect: the difference (top - bottom) is -2x^2 + 10x - 12, but the problem states the curves are y = -2x^2 + 11x - 9 and y
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer returned a non-object
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.