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Area between curves

Problem 5.12 · medium Video

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

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

Answer \( \frac{1}{3} \)

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

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 solution claims y = -2x^2 - 3x - 1 is on top, but for x in (-1, 0), the line y = -x - 1 is actually above the parabola. The integrand used (-2x^2 - 2x) is negative on this interval, so the integral yields -1/3, not 1/3. The area should be the absolute value.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution claims y = -2x^2 - 3x - 1 is on top, but for x in (-1, 0), the line y = -x - 1 is actually above the parabola. The integrand used (-2x^2 - 2x) is negative on this interval, so the integral yields -1/3, not 1/3. The area should be the absolute value.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution claims y = -2x^2 - 3x - 1 is on top, but for x in (-1, 0), the line y = -x - 1 is actually above the parabola. The integrand used (-2x^2 - 2x) is negative in this interval, so the integral yields -1/3, not 1/3. The area should be the absolute value.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution incorrectly identifies the top curve. For x in (-1, 0), -2x^2 - 3x - 1 is negative and below -x - 1, so the integrand should be reversed (bottom - top) or the absolute value taken. The computed integral of the stated 'top - bottom' yields -1/3, not 1/3.
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.