Owl: Area in polar coordinates is not a stack of vertical strips. From the origin, a tiny angle dθ cuts a thin sector. If the radius is about r, that sector's area is roughly ½ r² dθ — the same formula as a pizza slice.
That's why the squared radius appears. Integrating ∫ r dθ would be dimensionally and geometrically wrong for area.
We'll sketch regions, find honest θ bounds (especially for overlapping petals), and integrate ½ r². Traps: using ∫ r dθ, and botching petal limits so you double-count or miss a lobe.