Dynamic filling of a region of a polar plot

This is what you need:

Manipulate[ParametricPlot[
  r (1 + 2 Sin[θ]) {Cos[θ], Sin[θ]},
  {θ, 0, thmax},
  {r, 0, 1},
  PlotRange -> {{-2.25, 2.25}, {-0.5, 3.5}},
  PerformanceGoal -> "Quality"
  ], {thmax, 0.01, 2 Pi}]