eixos=ContourPlot[{x==0,y==0},{x,-0.5,4.5},{y,-0.5,4.5},ContourStyle->{{Black,Thick},{Black,Thick}}];

curvas=ParametricPlot[{{(3.5+Sin[-8θ+3π]) Cos[θ],(3.5+Sin[-8θ+3π]) Sin[θ]},{2 Cos[θ],2 Sin[θ]}},{θ,π/8,3π/8},PlotStyle->{{Blue,Thick},{Red,Thick}}];

radiais=ParametricPlot[{{r Cos[π/8],r Sin[π/8]},{r Cos[3π/8],r Sin[3π/8]}},{r,0,3.5},PlotStyle->{{Dashed,Black,Thick},{Dashed,Black,Thick}}];

region=ParametricPlot[{r Cos[θ],r Sin[θ]},{θ,π/8,3π/8},{r,2,3.5+Sin[-8θ+3π]},Mesh->True,PlotStyle->Opacity[0.2],PlotPoints->50];

Show[eixos,radiais,curvas,region,Graphics[{PointSize[Large],Black,Thick,Circle[{0,0},1.5,{0,π/8}],,Circle[{0,0},0.5,{0,3π/8}],Text[Style["α",Large],{1.7,0.3}],Text[Style["β",Large],{0.6,0.4}]}]]