Home›Calculus 2›Area in polar coordinates›Problem 8.19

Area in polar coordinates

Problem 8.19 · easy

Find the area of the region inside the cardioid \( \displaystyle r = 4 \cos{\left(\theta \right)} + 4 \).

Area in polar coordinates is ½∫ r² dθ over the angles that trace the region once.
Reviewed
Here θ runs from 0 to 2*pi.
Reviewed
\[ \int\limits_{0}^{2 \pi} \frac{\left(4 \cos{\left(\theta \right)} + 4\right)^{2}}{2}\, d\theta = 24 \pi \]
Integrate.✓ Proved

Answer \( 24 \pi \)

✓ Nihil obstat Lines: 1 proved, 2 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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	the shoelace formula on 20,000 points of the curve gives the same area

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the formula for area in polar coordinates, the correct bounds for a full traversal of the cardioid, and the integration result is verified.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the formula for area in polar coordinates, the correct bounds for a full traversal of the cardioid, and the integration result is verified.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the formula for area in polar coordinates and the appropriate bounds for a full trace of the cardioid. The integration result is correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The setup correctly identifies the polar area formula and the interval [0, 2π] required to trace the cardioid once. The integration result is correct.
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/polar_area, checked 2026-09-28 with SymPy 1.14.0.