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Area between curves

Problem 5.8 · medium Video

Find the area of the region bounded by \( \displaystyle y = - x^{2} + x + 1 \) and \( \displaystyle y = 3 x + 1 \).

\[ - x \left(x + 2\right) = - x^{2} - 2 x \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = -2 and x = 0; between them y = -x**2 + x + 1 is on top.
\[ \int\limits_{-2}^{0} \left(- x^{2} - 2 x\right)\, dx = \frac{4}{3} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{4}{3} \)

The curves y = -x² + x + 1 and y = 3x + 1 between x = -2 and x = 0. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 4/3.

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: inconclusive — reviewer returned a non-object

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer returned a non-object
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The intersection points are incorrect. Solving -x^2 + x + 1 = 3x + 1 yields x^2 + 2x = 0, so x = 0 and x = -2. However, the integrand used in step 1 is -x^2 - 2x, which corresponds to (-x^2 + x + 1) - (3x + 1) = -x^2 - 2x. Wait, let's recheck the subtraction: (-x^2 + x + 1) - (3x + 1) = -x^2 - 2x. This is correct. The roots of -x^2 - 2x = 0 are x(-x-2)=0, so x=0 and x=-2. The bounds are correct. The integral of -x^2 - 2x from -2 to 0 is [-x^3/3 - x^2] from -2 to 0 = 0 - ( -(-8)/3 - 4 ) = - (8/3 - 4) = - (8/3 - 12/3) = -(-4/3) = 4/3. The calculation is correct. Why did I think it was wrong? Let me re-read the prompt's solution carefully. Step 1: `-x*(x + 2), -x**2 - 2*x`. This is the integrand. Step 2: Cross at -2 and 0. Top is parabola. Step 3: Integral is 4/3. Actually, the solution is correct. Let me double check the intersection again. y1 = -x^2 + x + 1 y2 = 3x + 1 y1 = y2 => -x^2 + x + 1 = 3x + 1 => -x^2 - 2x = 0 => -x(x+2) = 0 => x=0, x=-2. Between -2 and 0, test x=-1: y1(-1) = -1 - 1 + 1 = -1. y2(-1) = -3 + 1 = -2. y1 > y2. So parabola is on top. Integrand: y1 - y2 = -x^2 - 2x. Integral from -2 to 0 of (-x^2 - 2x) dx. Antiderivative: -x^3/3 - x^2. At 0: 0. At -2: -(-8)/3 - 4 = 8/3 - 12/3 = -4/3. Result: 0 - (-4/3) = 4/3. The solution is mathematically sound. I will mark it as pass.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer returned a non-object
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.