Area between curves
Problem 5.193 · medium
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 5 x + 19 \) and \( \displaystyle y = x + 3 \).
- \[ \left(8 - 2 x\right) \left(x + 2\right) = - 2 x^{2} + 4 x + 16 \]Set the curves equal: top − bottom, factored.✓ Proved
- They cross at x = -2 and x = 4; between them y = -2*x**2 + 5*x + 19 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.
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 integrand in step 1 is incorrect; the difference between the curves is -2x^2 + 4x + 16, but the problem states the top curve is -2x^2 + 5x + 19. The correct difference is -2x^2 + 4x + 16 only if the bottom curve was x+3 and the top was -2x^2+5x+19? Wait. Top - Bottom = (-2x^2 + 5x + 19) - (x + 3) = -2x^2 + 4x + 16. The algebra in step 1 is actually correct for the difference. However, step 1 says '-2*(x - 4)*(x + 2) = -2*x**2 + 4*x + 16'. Let's check the roots of -2x^2 + 4x + 16 = 0. x^2 - 2x - 8 = 0 => (x-4)(x+2)=0. Roots are 4 and -2. This matches step 2. The integral of -2x^2 + 4x + 16 from -2 to 4 is indeed 72. So the math is correct. Why did I think it was wrong? Let me re-read carefully. Problem: y = -2x^2 + 5x + 19 and y = x + 3. Difference: -2x^2 + 4x + 16. Step 1 computes this difference. Step 2 finds roots. Step 3 integrates. The solution is correct. Wait, is there a subtle error? 'Set the curves equal: top - bottom, factored.' The equation shown is -2*(x-4)*(x+2) = ... which is the factored form of the difference. This is fine. The verdict should be pass.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The integrand in step 1 is incorrect; the difference between the curves is -2x^2 + 4x + 16, but the problem states the top curve is -2x^2 + 5x + 19. The correct difference is -2x^2 + 4x + 16 only if the bottom curve was x+3 and the top was -2x^2+5x+19? Wait. Top - Bottom = (-2x^2 + 5x + 19) - (x + 3) = -2x^2 + 4x + 16. The algebra in step 1 is actually correct for the difference. However, step 1 says '-2*(x - 4)*(x + 2) = -2*x**2 + 4*x + 16'. Let's check the roots of -2x^2 + 4x + 16 = 0. x^2 - 2x - 8 = 0 => (x-4)(x+2)=0. Roots are 4 and -2. This matches step 2. The integral of -2x^2 + 4x + 16 from -2 to 4 is indeed 72. So the math is correct. Why did I think it was wrong? Let me re-read carefully. Problem: y = -2x^2 + 5x + 19 and y = x + 3. Difference: -2x^2 + 4x + 16. Step 1 computes this difference. Step 2 finds roots. Step 3 integrates. The solution is correct. Wait, is there a subtle error? 'Set the curves equal: top - bottom, factored.' The equation shown is -2*(x-4)*(x+2) = ... which is the factored form of the difference. This is fine. The verdict should be 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 + 4x + 16, but the problem states the parabola is -2x^2 + 5x + 19. The correct difference is -2x^2 + 4x + 16 only if the linear term was 4x, but it is 5x. Wait, let me re-calculate. Top: -2x^2 + 5x + 19. Bottom: x + 3. Difference: (-2x^2 + 5x + 19) - (x + 3) = -2x^2 + 4x + 16. Okay, the algebra in line 1 is actually correct for the difference. Let me check the roots. -2(x-4)(x+2) = -2(x^2 - 2x - 8) = -2x^2 + 4x + 16. Correct. Let me check the intersection points. -2x^2 + 5x + 19 = x + 3 => -2x^2 + 4x + 16 = 0 => x^2 - 2x - 8 = 0 => (x-4)(x+2)=0. Roots are 4 and -2. Correct. Let me check the integral. Integral of -2x^2 + 4x + 16 from -2 to 4. 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 actually correct. Why did I think it was wrong? I misread the problem statement or the solution. Let me re-read carefully. Problem: y = -2x^2 + 5x + 19 and y = x + 3. Solution line 1: -2*(x-4)*(x+2) = -2x^2 + 4x + 16. This is the integrand. Is the integrand correct? Top - Bottom = (-2x^2 + 5x + 19) - (x + 3) = -2x^2 + 4x + 16. Yes. Roots: -2x^2 + 4x + 16 = 0 -> x^2 - 2x - 8 = 0 -> (x-4)(x+2)=0. Yes. Integral calculation: 72. Yes. The solution is sound.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.