Each curve completes one loop over the interval \(0\le t\le2\pi\). Find the intersections of the curves within this interval.
\(6-6\sin\theta=6\implies 1-\sin\theta=1\implies \sin\theta=0\implies \theta=0,\theta=\pi\)
The region of interest has an area given by the double integral
\(\displaystyle\int_\pi^{2\pi}\int_6^{6-6\sin\theta}r\,\mathrm dr\,\mathrm d\theta\)
equivalent to the single integral
\(\displaystyle\frac12\int_\pi^{2\pi}\bigg((6-6\sin\theta)^2-6^2\bigg)\,\mathrm d\theta\)
which evaluates to \(9\pi+72\).
