Area in polar coordinates
Problem 8.25 · easy
Find the area of one petal of the rose \( \displaystyle r = 2 \cos{\left(2 \theta \right)} \).
- Area in polar coordinates is ½∫ r² dθ over the angles that trace the region once.
- Here θ runs from -pi/4 to pi/4.
- \[ \int\limits_{- \frac{\pi}{4}}^{\frac{\pi}{4}} 2 \cos^{2}{\left(2 \theta \right)}\, d\theta = \frac{\pi}{2} \]Integrate.✓ Proved
Answer \( \frac{\pi}{2} \)
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.