Area between curves
Problem 5.122 · medium
Find the area of the region bounded by \( \displaystyle y = - x^{2} - 4 x - 3 \) and \( \displaystyle y = - 3 x - 3 \).
- \[ - x \left(x + 1\right) = - x^{2} - x \]Set the curves equal: top − bottom, factored.✓ Proved
- They cross at x = -1 and x = 0; between them y = -x**2 - 4*x - 3 is on top.
- \[ \int\limits_{-1}^{0} \left(- x^{2} - x\right)\, dx = \frac{1}{6} \]Area = ∫ (top − bottom) dx.✓ Proved
Answer \( \frac{1}{6} \)
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 intersection points are incorrect; solving -x^2 - 4x - 3 = -3x - 3 yields x = -1 and x = 0, but the difference function is -x^2 - x, which is negative on (-1, 0). The parabola is actually below the line in this interval, so the integrand should be reversed or absolute value taken, though the magnitude happens to be correct by coincidence of symmetry, the statement 'parabola is on top' is false.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The intersection points are incorrect; solving -x^2 - 4x - 3 = -3x - 3 yields x = -1 and x = 0, but the difference function is -x^2 - x, which is negative on (-1, 0). The parabola is actually below the line in this interval, so the integrand should be reversed or absolute value taken, though the magnitude happens to be correct by coincidence of symmetry, the statement 'parabola is on top' is false.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-27 — The intersection points are incorrect; solving -x^2 - 4x - 3 = -3x - 3 yields x = 0 and x = -1 is wrong, the correct roots are x = 0 and x = -1? Wait, -x^2 - x = 0 => x(x+1)=0, so x=0, -1. Let's recheck the intersection. -x^2 - 4x - 3 = -3x - 3 => -x^2 - x = 0 => x^2 + x = 0 => x(x+1)=0. So x=0 and x=-1 are correct. However, the integrand in step 1 is -x^2 - x. Let's check top - bottom. Top is parabola? At x = -0.5, y_p = -0.25 + 2 - 3 = -1.25. y_l = 1.5 - 3 = -1.5. Parabola is higher. So top - bottom = (-x^2 - 4x - 3) - (-3x - 3) = -x^2 - x. This is correct. The integral of -x^2 - x from -1 to 0 is [-x^3/3 - x^2/2] from -1 to 0 = 0 - (1/3 - 1/2) = 1/6. The solution is actually correct. Why did I think it was wrong? Let me re-read carefully. Step 1 says 'Set the curves equal: top - bottom, factored'. The equation shown is '-x*(x + 1), -x**2 - x'. This is the integrand. Step 2 says they cross at -1 and 0. Step 3 integrates. The logic holds. Is there a subtle error? 'top - bottom' is positive on (-1, 0)? -x^2 - x = -x(x+1). For x in (-1, 0), x is negative, x+1 is positive, so -x is positive, product is positive. Yes. The area is 1/6. The solution is sound.gpt-oss:20b: pass 2026-09-27
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 2026-09-27 with SymPy 1.14.0.