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Area between curves, integrating in y

Problem 5.557 · medium

Find the area of the region between \( \displaystyle x = y^{2} \) and \( \displaystyle x = 8 - y^{2} \).

These are functions of y, so slice horizontally: a strip at height y runs from the left curve to the right one.
Reviewed
\[ \left. 8 - 2 y^{2} \right|_{\substack{ y=-2 }} = 0 \]
The curves meet at y = -2 ...✓ Proved
\[ \left. 8 - 2 y^{2} \right|_{\substack{ y=2 }} = 0 \]
... and at y = 2.✓ Proved
\[ \int\limits_{-2}^{2} \left(8 - 2 y^{2}\right)\, dy = \frac{64}{3} \]
A = ∫ (right − left) dy.✓ Proved

Answer \( \frac{64}{3} \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ 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	the shoelace formula on 8,000 points around the region's boundary

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the region, sets up the horizontal slicing integral with the correct bounds and integrand, and computes the result accurately.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the region, sets up the horizontal slicing integral with the correct bounds and integrand, and computes the result accurately.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the integration variable and limits, and the integrand represents the correct horizontal width (right curve minus left curve). The final result is correct.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14
gpt-oss:20b: pass 2026-10-14

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_y, checked 2026-10-17 with SymPy 1.14.0.