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