Area between curves
Problem 5.70 · medium Video
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + x + 15 \) and \( \displaystyle y = - 3 x - 1 \).
- \[ \left(8 - 2 x\right) \left(x + 2\right) = - 2 x^{2} + 4 x + 16 \]Set the curves equal: top − bottom, factored.✓ Proved
- They cross at x = -2 and x = 4; between them y = -2*x**2 + x + 15 is on top.
- \[ \int\limits_{-2}^{4} \left(- 2 x^{2} + 4 x + 16\right)\, dx = 72 \]Area = ∫ (top − bottom) dx.✓ Proved
Answer \( 72 \)
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: passqwen3.6:27b-mlx: fail (error) — The setup in step 1 is incorrect: the difference (top - bottom) is -2x^2 + 4x + 16, but the problem states the top curve is -2x^2 + x + 15. The correct difference should be -2x^2 + 4x + 16 only if the bottom curve was -3x - 1 and the top was -2x^2 + 4x + 16, which contradicts the given top curve. Wait, let's re-calculate: (-2x^2 + x + 15) - (-3x - 1) = -2x^2 + 4x + 16. The algebra in step 1 is actually correct for the difference. However, step 1 says '-2*(x - 4)*(x + 2)' which expands to -2(x^2 - 2x - 8) = -2x^2 + 4x + 16. This matches the difference. So the algebra is correct. Let's re-read carefully. Step 1 says 'Set the curves equal: top - bottom, factored.' The equation shown is '-2*(x - 4)*(x + 2), -2*x**2 + 4*x + 16'. This implies the integrand is -2x^2 + 4x + 16. The roots of -2x^2 + 4x + 16 = 0 are x = -2 and x = 4. This matches step 2. The integral of -2x^2 + 4x + 16 from -2 to 4 is indeed 72. The solution is actually correct. Why did I think it was wrong? Let me double check the problem statement. y = -2x^2 + x + 15 and y = -3x - 1. Difference: -2x^2 + x + 15 - (-3x - 1) = -2x^2 + 4x + 16. Roots: -2x^2 + 4x + 16 = 0 => x^2 - 2x - 8 = 0 => (x-4)(x+2)=0. Roots are 4 and -2. Integral from -2 to 4 of (-2x^2 + 4x + 16) dx. Antiderivative: -2/3 x^3 + 2x^2 + 16x. Evaluated at 4: -2/3(64) + 2(16) + 16(4) = -128/3 + 32 + 64 = -128/3 + 96 = (-128 + 288)/3 = 160/3. Evaluated at -2: -2/3(-8) + 2(4) + 16(-2) = 16/3 + 8 - 32 = 16/3 - 24 = (16 - 72)/3 = -56/3. Result: 160/3 - (-56/3) = 216/3 = 72. The solution is correct. The text in step 1 is slightly confusingly phrased ('Set the curves equal' usually implies finding intersections, but it shows the difference), but the math is sound. Step 2 correctly identifies the bounds and the top curve. Step 3 correctly integrates. I will mark it as pass.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The setup in step 1 is incorrect: the difference (top - bottom) is -2x^2 + 4x + 16, but the problem states the top curve is -2x^2 + x + 15. The correct difference should be -2x^2 + 4x + 16 only if the bottom curve was -3x - 1 and the top was -2x^2 + 4x + 16, which contradicts the given top curve. Wait, let's re-calculate: (-2x^2 + x + 15) - (-3x - 1) = -2x^2 + 4x + 16. The algebra in step 1 is actually correct for the difference. However, step 1 says '-2*(x - 4)*(x + 2)' which expands to -2(x^2 - 2x - 8) = -2x^2 + 4x + 16. This matches the difference. So the algebra is correct. Let's re-read carefully. Step 1 says 'Set the curves equal: top - bottom, factored.' The equation shown is '-2*(x - 4)*(x + 2), -2*x**2 + 4*x + 16'. This implies the integrand is -2x^2 + 4x + 16. The roots of -2x^2 + 4x + 16 = 0 are x = -2 and x = 4. This matches step 2. The integral of -2x^2 + 4x + 16 from -2 to 4 is indeed 72. The solution is actually correct. Why did I think it was wrong? Let me double check the problem statement. y = -2x^2 + x + 15 and y = -3x - 1. Difference: -2x^2 + x + 15 - (-3x - 1) = -2x^2 + 4x + 16. Roots: -2x^2 + 4x + 16 = 0 => x^2 - 2x - 8 = 0 => (x-4)(x+2)=0. Roots are 4 and -2. Integral from -2 to 4 of (-2x^2 + 4x + 16) dx. Antiderivative: -2/3 x^3 + 2x^2 + 16x. Evaluated at 4: -2/3(64) + 2(16) + 16(4) = -128/3 + 32 + 64 = -128/3 + 96 = (-128 + 288)/3 = 160/3. Evaluated at -2: -2/3(-8) + 2(4) + 16(-2) = 16/3 + 8 - 32 = 16/3 - 24 = (16 - 72)/3 = -56/3. Result: 160/3 - (-56/3) = 216/3 = 72. The solution is correct. The text in step 1 is slightly confusingly phrased ('Set the curves equal' usually implies finding intersections, but it shows the difference), but the math is sound. Step 2 correctly identifies the bounds and the top curve. Step 3 correctly integrates. I will mark it as pass.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-27 — The setup in line 1 incorrectly defines the integrand as -2x^2 + 4x + 16, which corresponds to the difference between y = -2x^2 + 4x + 16 and y = -3x - 1, rather than the problem's y = -2x^2 + x + 15. Although the intersection points and final answer happen to be correct for the stated problem, the intermediate algebraic model is wrong.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the intersection points, determines the correct integrand (top curve minus bottom curve), and computes the definite integral accurately.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.