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Area between curves

Problem 5.13 · medium Video

Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 13 x - 11 \) and \( \displaystyle y = 1 - 2 x \).

\[ \left(12 - 3 x\right) \left(x - 1\right) = - 3 x^{2} + 15 x - 12 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = 1 and x = 4; between them y = -3*x**2 + 13*x - 11 is on top.
\[ \int\limits_{1}^{4} \left(- 3 x^{2} + 15 x - 12\right)\, dx = \frac{27}{2} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{27}{2} \)

The curves y = -3x² + 13x - 11 and y = 1 - 2x between x = 1 and x = 4. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 27/2.

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 intersection points are incorrect; the curves intersect at x = 2 and x = 5, not x = 1 and x = 4. Consequently, the integrand used (-3x^2 + 15x - 12) does not match the difference of the given functions (-3x^2 + 15x - 12 vs -3x^2 + 15x - 12? Wait, let's check: (-3x^2+13x-11) - (1-2x) = -3x^2 + 15x - 12. The integrand is actually correct for the difference, but the bounds are wrong. The roots of -3x^2+15x-12=0 are x=1 and x=4? Let's check: -3(1)^2+15(1)-12 = 0. -3(4)^2+15(4)-12 = -48+60-12=0. So the integrand -3x^2+15x-12 has roots 1 and 4. But do the original curves intersect at 1 and 4? y1(1) = -3+13-11 = -1. y2(1) = 1-2 = -1. Yes. y1(4) = -3(16)+13(4)-11 = -48+52-11 = -7. y2(4) = 1-8 = -7. Yes. So the intersection points ARE 1 and 4. My initial check was wrong. Let me re-verify. Original curves: y = -3x^2 + 13x - 11 and y = 1 - 2x. Difference: (-3x^2 + 13x - 11) - (1 - 2x) = -3x^2 + 15x - 12. Roots of -3x^2 + 15x - 12 = 0 => 3x^2 - 15x + 12 = 0 => x^2 - 5x + 4 = 0 => (x-1)(x-4)=0. Roots are 1 and 4. So the bounds 1 and 4 are correct. The integrand is correct. The integral of -3x^2 + 15x - 12 from 1 to 4 is indeed 27/2. Wait, did I make a mistake in my thought process? Let's re-read the prompt's solution. 1. Equation: -3*(x-4)*(x-1) expands to -3(x^2-5x+4) = -3x^2+15x-12. This matches the difference. 2. Sentence: Cross at 1 and 4. Correct. Top is parabola. Correct. 3. Equation: Integral is 27/2. Correct. So the solution is actually correct. Why did I think it was wrong? I must have miscalculated the intersection in my head initially. Let's double check y(2) and y(5) just in case. If x=2: y1 = -12+26-11=3. y2 = 1-4=-3. Not equal. If x=5: y1 = -75+65-11=-21. y2 = 1-10=-9. Not equal. Okay, the solution is correct. The verdict should be pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The intersection points are incorrect; the curves intersect at x = 2 and x = 5, not x = 1 and x = 4. Consequently, the integrand used (-3x^2 + 15x - 12) does not match the difference of the given functions (-3x^2 + 15x - 12 vs -3x^2 + 15x - 12? Wait, let's check: (-3x^2+13x-11) - (1-2x) = -3x^2 + 15x - 12. The integrand is actually correct for the difference, but the bounds are wrong. The roots of -3x^2+15x-12=0 are x=1 and x=4? Let's check: -3(1)^2+15(1)-12 = 0. -3(4)^2+15(4)-12 = -48+60-12=0. So the integrand -3x^2+15x-12 has roots 1 and 4. But do the original curves intersect at 1 and 4? y1(1) = -3+13-11 = -1. y2(1) = 1-2 = -1. Yes. y1(4) = -3(16)+13(4)-11 = -48+52-11 = -7. y2(4) = 1-8 = -7. Yes. So the intersection points ARE 1 and 4. My initial check was wrong. Let me re-verify. Original curves: y = -3x^2 + 13x - 11 and y = 1 - 2x. Difference: (-3x^2 + 13x - 11) - (1 - 2x) = -3x^2 + 15x - 12. Roots of -3x^2 + 15x - 12 = 0 => 3x^2 - 15x + 12 = 0 => x^2 - 5x + 4 = 0 => (x-1)(x-4)=0. Roots are 1 and 4. So the bounds 1 and 4 are correct. The integrand is correct. The integral of -3x^2 + 15x - 12 from 1 to 4 is indeed 27/2. Wait, did I make a mistake in my thought process? Let's re-read the prompt's solution. 1. Equation: -3*(x-4)*(x-1) expands to -3(x^2-5x+4) = -3x^2+15x-12. This matches the difference. 2. Sentence: Cross at 1 and 4. Correct. Top is parabola. Correct. 3. Equation: Integral is 27/2. Correct. So the solution is actually correct. Why did I think it was wrong? I must have miscalculated the intersection in my head initially. Let's double check y(2) and y(5) just in case. If x=2: y1 = -12+26-11=3. y2 = 1-4=-3. Not equal. If x=5: y1 = -75+65-11=-21. y2 = 1-10=-9. Not equal. Okay, the solution is correct. The verdict should be pass.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The intersection points are incorrect; solving -3x^2 + 13x - 11 = 1 - 2x yields x = 1 and x = 4/3, not x = 4. Consequently, the integration bounds and the resulting area are wrong.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The intersection points are incorrect; solving -3x^2 + 13x - 11 = 1 - 2x yields x = 1 and x = 10/3, not x = 4. Consequently, the integration bounds and the integrand (which uses the wrong difference of functions) are both wrong.
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.