Area between curves
Problem 5.197 · medium
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 13 x - 11 \) and \( \displaystyle y = 3 x + 1 \).
- \[ \left(6 - 2 x\right) \left(x - 2\right) = - 2 x^{2} + 10 x - 12 \]Set the curves equal: top − bottom, factored.✓ Proved
- They cross at x = 2 and x = 3; between them y = -2*x**2 + 13*x - 11 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 \( \frac{1}{3} \)
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: passqwen3.6:27b-mlx: fail (error) — The setup in line 1 is incorrect; it factors -2x^2 + 10x - 12, but the actual integrand (top - bottom) is -2x^2 + 10x - 12, which does factor as -2(x-2)(x-3). However, line 2 claims the curves cross at x=2 and x=3. Let's check: -2(2)^2 + 13(2) - 11 = -8 + 26 - 11 = 7. 3(2) + 1 = 7. Correct. -2(3)^2 + 13(3) - 11 = -18 + 39 - 11 = 10. 3(3) + 1 = 10. Correct. So the bounds are correct. The integrand is (-2x^2 + 13x - 11) - (3x + 1) = -2x^2 + 10x - 12. This factors to -2(x^2 - 5x + 6) = -2(x-2)(x-3). Line 1 says -2*(x-3)*(x-2) = -2*x**2 + 10*x - 12. This is algebraically correct. Line 3 integrates this from 2 to 3. The integral of -2(x-2)(x-3) from 2 to 3 is indeed 1/3. Wait, let me re-read carefully. Line 1 says "Set the curves equal: top - bottom, factored." The equation shown is -2*(x-3)*(x-2) = -2*x**2 + 10*x - 12. This is an identity, not the setup of the integral. But the comment says it's the top-bottom. The actual top-bottom is -2x^2 + 10x - 12. So the integrand is correct. The bounds are correct. The result is correct. Why did I think it was wrong? Let me check the problem statement again. y = -2x^2 + 13x - 11 and y = 3x + 1. Intersection: -2x^2 + 10x - 12 = 0 => x^2 - 5x + 6 = 0 => (x-2)(x-3)=0. x=2, 3. Integral from 2 to 3 of (-2x^2 + 10x - 12) dx. Antiderivative: -2/3 x^3 + 5x^2 - 12x. Evaluated at 3: -2/3(27) + 5(9) - 36 = -18 + 45 - 36 = -9. Evaluated at 2: -2/3(8) + 5(4) - 24 = -16/3 + 20 - 24 = -16/3 - 4 = -28/3. Difference: -9 - (-28/3) = -27/3 + 28/3 = 1/3. The solution is correct. Is there a subtle error? Line 1 is marked [equation, proved]. It shows an algebraic identity. Line 2 states the bounds and which is on top. Line 3 computes the integral. Everything seems correct. Wait, look at line 1 again. "-2*(x - 3)*(x - 2) = -2*x**2 + 10*x - 12". This is just expanding the factored form. It doesn't explicitly state that this expression IS the top minus bottom, though the comment says so. Line 2 says "between them y = ... is on top". It doesn't explicitly write the integrand, but Line 3 uses the correct integrand. Is there a misleading part? "Set the curves equal: top - bottom, factored." This is a bit garbled. Setting curves equal finds intersections. Top - bottom finds the integrand. The line shows the expansion of the factored integrand. It's a bit sloppy but not wrong. Let me look closer. Is the area 1/3? Yes. Is the setup correct? Yes. I will mark it pass.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The setup in line 1 is incorrect; it factors -2x^2 + 10x - 12, but the actual integrand (top - bottom) is -2x^2 + 10x - 12, which does factor as -2(x-2)(x-3). However, line 2 claims the curves cross at x=2 and x=3. Let's check: -2(2)^2 + 13(2) - 11 = -8 + 26 - 11 = 7. 3(2) + 1 = 7. Correct. -2(3)^2 + 13(3) - 11 = -18 + 39 - 11 = 10. 3(3) + 1 = 10. Correct. So the bounds are correct. The integrand is (-2x^2 + 13x - 11) - (3x + 1) = -2x^2 + 10x - 12. This factors to -2(x^2 - 5x + 6) = -2(x-2)(x-3). Line 1 says -2*(x-3)*(x-2) = -2*x**2 + 10*x - 12. This is algebraically correct. Line 3 integrates this from 2 to 3. The integral of -2(x-2)(x-3) from 2 to 3 is indeed 1/3. Wait, let me re-read carefully. Line 1 says "Set the curves equal: top - bottom, factored." The equation shown is -2*(x-3)*(x-2) = -2*x**2 + 10*x - 12. This is an identity, not the setup of the integral. But the comment says it's the top-bottom. The actual top-bottom is -2x^2 + 10x - 12. So the integrand is correct. The bounds are correct. The result is correct. Why did I think it was wrong? Let me check the problem statement again. y = -2x^2 + 13x - 11 and y = 3x + 1. Intersection: -2x^2 + 10x - 12 = 0 => x^2 - 5x + 6 = 0 => (x-2)(x-3)=0. x=2, 3. Integral from 2 to 3 of (-2x^2 + 10x - 12) dx. Antiderivative: -2/3 x^3 + 5x^2 - 12x. Evaluated at 3: -2/3(27) + 5(9) - 36 = -18 + 45 - 36 = -9. Evaluated at 2: -2/3(8) + 5(4) - 24 = -16/3 + 20 - 24 = -16/3 - 4 = -28/3. Difference: -9 - (-28/3) = -27/3 + 28/3 = 1/3. The solution is correct. Is there a subtle error? Line 1 is marked [equation, proved]. It shows an algebraic identity. Line 2 states the bounds and which is on top. Line 3 computes the integral. Everything seems correct. Wait, look at line 1 again. "-2*(x - 3)*(x - 2) = -2*x**2 + 10*x - 12". This is just expanding the factored form. It doesn't explicitly state that this expression IS the top minus bottom, though the comment says so. Line 2 says "between them y = ... is on top". It doesn't explicitly write the integrand, but Line 3 uses the correct integrand. Is there a misleading part? "Set the curves equal: top - bottom, factored." This is a bit garbled. Setting curves equal finds intersections. Top - bottom finds the integrand. The line shows the expansion of the factored integrand. It's a bit sloppy but not wrong. Let me look closer. Is the area 1/3? Yes. Is the setup correct? Yes. I will mark it pass.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-10-03 — The setup in line 1 is incorrect; the difference between the curves is -2x^2 + 10x - 12, but the problem states the parabola is -2x^2 + 13x - 11. The solution solves for a different problem entirely.gpt-oss:20b: pass 2026-10-03
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 2026-10-03 with SymPy 1.14.0.