Home›Calculus 1›Area between curves›Problem 5.142

Area between curves

Problem 5.142 · medium Video

Find the area of the region bounded by \( \displaystyle y = - x^{2} - 2 x + 7 \) and \( \displaystyle y = 1 - x \).

\[ \left(2 - x\right) \left(x + 3\right) = - x^{2} - x + 6 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = -3 and x = 2; between them y = -x**2 - 2*x + 7 is on top.
\[ \int\limits_{-3}^{2} \left(- x^{2} - x + 6\right)\, dx = \frac{125}{6} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{125}{6} \)

The curves y = -x² - 2x + 7 and y = 1 - x between x = -3 and x = 2. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 125/6.

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. 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 is incorrect: it factors the difference as -(x-2)(x+3), which equals -x^2 - x + 6, but the actual difference (top - bottom) is -x^2 - x + 8. Consequently, the integration bounds and integrand do not match the problem statement, leading to an incorrect area.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — The setup in line 1 is incorrect: it factors the difference as -(x-2)(x+3), which equals -x^2 - x + 6, but the actual difference (top - bottom) is -x^2 - x + 8. Consequently, the integration bounds and integrand do not match the problem statement, leading to an incorrect area.
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, and computes the definite integral accurately.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the intersection points and the relative position of the curves. The integrand represents the correct difference (top minus bottom), and the integration yields the correct area.
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.