A thin polar slice is a tiny sector, not a rectangle.
A sector of radius r and angle Δθ has area ½r²Δθ. Summing sectors and taking the limit produces the polar area integral. For one petal of r=cos(3θ), integrate from −π/6 to π/6. Using cos²(3θ)=(1+cos(6θ))/2 yields π/12.
Make sure the parameter interval traces the intended region once. A longer interval may count the same area repeatedly.
For r = cos(3θ), this petal lies between −π/6 and π/6. Each infinitesimal sector contributes ½r² dθ; radius is nonnegative on this interval.