Home›Calculus 1›Area between curves, integrating in y›Problem 5.559

Area between curves, integrating in y

Problem 5.559 · medium

Find the area of the region between \( \displaystyle x = y^{2} \) and \( \displaystyle x = y + 6 \).

These are functions of y, so slice horizontally: a strip at height y runs from the left curve to the right one.
Reviewed
\[ \left. - y^{2} + y + 6 \right|_{\substack{ y=-2 }} = 0 \]
The curves meet at y = -2 ...✓ Proved
\[ \left. - y^{2} + y + 6 \right|_{\substack{ y=3 }} = 0 \]
... and at y = 3.✓ Proved
\[ \int\limits_{-2}^{3} \left(- y^{2} + y + 6\right)\, dy = \frac{125}{6} \]
A = ∫ (right − left) dy.✓ Proved

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

✓ 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 integration variable and bounds, and the setup for the area integral is sound.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the integration variable and bounds, and the setup for the area integral is sound.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the curves, determines the intersection points, sets up the integral with the correct integrand (right curve minus left curve), and computes the area accurately.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the curves as functions of y, determines the correct intersection points, and sets up the integral with the correct integrand (right curve minus left curve) and bounds.
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.