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}]}]]

region2=ParametricPlot[{r Cos[θ],r Sin[θ]},{θ,π/8,3π/8},{r,2,4},Mesh->True,PlotStyle->Opacity[0.2],PlotPoints->50,BoundaryStyle->{Black,Thick}];

Show[eixos,radiais,region2,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}],Text[Style["x",Large],{4.4,0.2}],Text[Style["y",Large],{0.2,4.4}]}]]

r[θ_]=1−Cos[θ] Sin[3 θ];

ParametricPlot[{r[θ]Cos[θ],-r[θ]Sin[θ]},{θ,0,2π}]

Plot[r[θ],{θ,0,2π}]

FindRoot[r'[θ],{θ,3.6}]

θmin=%[[1,2]];

rmin=r[%]

Plot[r[θ]-rmin,{θ,0,2π}]

ParametricPlot[{(r[θ]-rmin)Cos[θ],-(r[θ]-rmin)Sin[θ]},{θ,0,2π}]

newcurva=ParametricPlot[{1.5,2}+{-(r[θ]-rmin)Cos[θ+π/6],(r[θ]-rmin)Sin[θ+π/6]},{θ,θmin-π,θmin},PlotStyle->{Blue,Thick}]

Show[eixos,radiais,region2,newcurva,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}],Text[Style["x",Large],{4.4,0.2}],Text[Style["y",Large],{0.2,4.4}],Text[Style["r = a",Bold],{2.4,0.4}],Line[{{2,0.7},{2.3,0.5}}],Text[Style["r = b",Bold],{4,1}],Line[{{3.8,1.5},{4,1.2}}],Line[{{3.3,2},{4,2}}],Text[Style["R",Large],{4.2,2}]}]]