Area in polar coordinates
Problem 8.17 · easy
Find the area of the region inside \( \displaystyle r = 6 \sin{\left(\theta \right)} \).
- Area in polar coordinates is ½∫ r² dθ over the angles that trace the region once.
- Here θ runs from 0 to pi.
- \[ \int\limits_{0}^{\pi} 18 \sin^{2}{\left(\theta \right)}\, d\theta = 9 \pi \]Integrate.✓ Proved
Answer \( 9 \pi \)
Lines: 1 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. The explanation has not been reviewed yet.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 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 | the shoelace formula on 20,000 points of the curve gives the same area |
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 2026-09-26 with SymPy 1.14.0.