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Area between curves

Problem 5.149 · medium Video

Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 5 x + 10 \) and \( \displaystyle y = 2 - x \).

\[ \left(8 - 2 x\right) \left(x + 1\right) = - 2 x^{2} + 6 x + 8 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = -1 and x = 4; between them y = -2*x**2 + 5*x + 10 is on top.
\[ \int\limits_{-1}^{4} \left(- 2 x^{2} + 6 x + 8\right)\, dx = \frac{125}{3} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{125}{3} \)

The curves y = -2x² + 5x + 10 and y = 2 - x between x = -1 and x = 4. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 125/3.

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 calculates the difference as -2x^2 + 6x + 8, but the actual difference (top - bottom) is (-2x^2 + 5x + 10) - (2 - x) = -2x^2 + 6x + 8. Wait, let me re-calculate. Top: -2x^2 + 5x + 10. Bottom: 2 - x. Difference: -2x^2 + 5x + 10 - 2 + x = -2x^2 + 6x + 8. The algebra in line 1 is correct. The roots of -2x^2 + 6x + 8 = 0 are x = 4 and x = -1. The integral of -2x^2 + 6x + 8 from -1 to 4 is indeed 125/3. Why did I think it was wrong? Let me re-read the prompt carefully. Problem: y = -2x^2 + 5x + 10 and y = 2 - x. Difference: (-2x^2 + 5x + 10) - (2 - x) = -2x^2 + 6x + 8. Roots: -2(x^2 - 3x - 4) = -2(x-4)(x+1). Roots at 4, -1. Integral: [-2/3 x^3 + 3x^2 + 8x] from -1 to 4. At 4: -2/3(64) + 3(16) + 32 = -128/3 + 48 + 32 = -128/3 + 80 = (-128 + 240)/3 = 112/3. At -1: -2/3(-1) + 3(1) - 8 = 2/3 + 3 - 8 = 2/3 - 5 = (2 - 15)/3 = -13/3. Result: 112/3 - (-13/3) = 125/3. The solution is actually correct. I was hallucinating an error. Let me double check the 'top' claim. At x=0, y1=10, y2=2. 10 > 2. So parabola is on top. Correct. Wait, looking at line 1 again: `-2*(x - 4)*(x + 1) = -2*x**2 + 6*x + 8`. This is the factored form of the integrand. The comment says 'Set the curves equal: top - bottom, factored.' This is slightly confusing phrasing but mathematically sound. The sentence in line 2 identifies the bounds and the top curve correctly. Line 3 computes the integral correctly. Is there any subtle error? 'y = -2x^2 + 5x + 10' and 'y = 2 - x'. Difference: -2x^2 + 6x + 8. Roots: -1, 4. Integral value: 125/3. It seems correct. Why would I fail it? Let me look really closely at the text. Line 1: `[equation, proved] -2*(x - 4)*(x + 1) = -2*x**2 + 6*x + 8 -- Set the curves equal: top − bottom, factored.` The equation itself is an identity. The comment explains that the RHS is `top - bottom`. This is correct. Line 2: `They cross at x = -1 and x = 4; between them y = -2*x**2 + 5*x + 10 is on top.` Correct. Line 3: `Integral(-2*x**2 + 6*x + 8, (x, -1, 4)) = 125/3`. Correct. Okay, I will pass it.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — The setup in line 1 calculates the difference as -2x^2 + 6x + 8, but the actual difference (top - bottom) is (-2x^2 + 5x + 10) - (2 - x) = -2x^2 + 6x + 8. Wait, let me re-calculate. Top: -2x^2 + 5x + 10. Bottom: 2 - x. Difference: -2x^2 + 5x + 10 - 2 + x = -2x^2 + 6x + 8. The algebra in line 1 is correct. The roots of -2x^2 + 6x + 8 = 0 are x = 4 and x = -1. The integral of -2x^2 + 6x + 8 from -1 to 4 is indeed 125/3. Why did I think it was wrong? Let me re-read the prompt carefully. Problem: y = -2x^2 + 5x + 10 and y = 2 - x. Difference: (-2x^2 + 5x + 10) - (2 - x) = -2x^2 + 6x + 8. Roots: -2(x^2 - 3x - 4) = -2(x-4)(x+1). Roots at 4, -1. Integral: [-2/3 x^3 + 3x^2 + 8x] from -1 to 4. At 4: -2/3(64) + 3(16) + 32 = -128/3 + 48 + 32 = -128/3 + 80 = (-128 + 240)/3 = 112/3. At -1: -2/3(-1) + 3(1) - 8 = 2/3 + 3 - 8 = 2/3 - 5 = (2 - 15)/3 = -13/3. Result: 112/3 - (-13/3) = 125/3. The solution is actually correct. I was hallucinating an error. Let me double check the 'top' claim. At x=0, y1=10, y2=2. 10 > 2. So parabola is on top. Correct. Wait, looking at line 1 again: `-2*(x - 4)*(x + 1) = -2*x**2 + 6*x + 8`. This is the factored form of the integrand. The comment says 'Set the curves equal: top - bottom, factored.' This is slightly confusing phrasing but mathematically sound. The sentence in line 2 identifies the bounds and the top curve correctly. Line 3 computes the integral correctly. Is there any subtle error? 'y = -2x^2 + 5x + 10' and 'y = 2 - x'. Difference: -2x^2 + 6x + 8. Roots: -1, 4. Integral value: 125/3. It seems correct. Why would I fail it? Let me look really closely at the text. Line 1: `[equation, proved] -2*(x - 4)*(x + 1) = -2*x**2 + 6*x + 8 -- Set the curves equal: top − bottom, factored.` The equation itself is an identity. The comment explains that the RHS is `top - bottom`. This is correct. Line 2: `They cross at x = -1 and x = 4; between them y = -2*x**2 + 5*x + 10 is on top.` Correct. Line 3: `Integral(-2*x**2 + 6*x + 8, (x, -1, 4)) = 125/3`. Correct. Okay, I will pass it.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the intersection points, determines the correct upper and lower functions on the interval, and computes the definite integral accurately.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The setup in step 1 incorrectly calculates the difference between the curves. The actual difference is (-2x^2 + 5x + 10) - (2 - x) = -2x^2 + 6x + 8, but the factorization provided (-2*(x - 4)*(x + 1)) expands to -2x^2 + 6x + 8, which matches the difference, not the original top curve. However, the roots of -2x^2 + 6x + 8 are indeed -1 and 4. Wait, let's re-read carefully. Step 1 says 'Set the curves equal: top - bottom, factored'. The expression -2*x**2 + 6*x + 8 is indeed the integrand. The factorization -2*(x-4)*(x+1) is correct for that integrand. The roots are -1 and 4. Step 2 claims the curves cross at -1 and 4. This is correct. Step 2 claims the quadratic is on top. This is correct for a downward parabola between roots. Step 3 integrates the correct difference. The solution is actually correct. Why did I think it was wrong? Let me re-verify the factorization. -2(x-4)(x+1) = -2(x^2 - 3x - 4) = -2x^2 + 6x + 8. The difference is (-2x^2 + 5x + 10) - (2 - x) = -2x^2 + 6x + 8. Everything is correct. The verdict should be pass.
gpt-oss:20b: pass 2026-09-28

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.