You can use ParametricRegion and overlay it on your original plot using Show:
Show[
PolarPlot[{θ, Exp[θ]}, {θ, 1, 1.2}, Frame ->True, PlotLegends -> "Expressions"]
, Region[ParametricRegion[(w*θ + (1 - w)* Exp[θ]) {Cos[θ], Sin[θ]}, {{θ, 1, 1.2}, {w, 0, 1}}]]
]
