Answer by Bob Hanlon for Area of ​a parametrically defined function

$Version(* "13.3.1 for Mac OS X ARM (64-bit) (July 24, 2023)" *)Clear["Global`*"]α = 1.525;β = 3.95;χ = 50;Hr = 58.3;He = 36.6;Bmax = 1.5;{x, y} = {α Sinh[Bmax Sin[γ] β] +    Bmax Sin[γ] χ,   (Bmax^2 - (Bmax Sin[            ArcTan[Hr/              Sqrt[He^2 + ((α Sinh[                    Bmax Sin[γ] β] +                    Bmax Sin[γ] χ))^2]]            ])^2)^(1/2) Sin[γ] -    (Bmax Sin[        ArcTan[         Hr/Sqrt[He^2 + ((α Sinh[Bmax Sin[γ] β] +                Bmax Sin[γ] χ))^2]]        ]) Cos[γ]};plt = ParametricPlot[{x, y}, {γ, -π, π},   PlotStyle -> Red,   AspectRatio -> 1/2];poly = Polygon[Cases[plt, Line[pts_] :> pts, Infinity]];Show[plt, Region[Style[poly, Opacity[0.5, LightBlue]]]]

Area[poly](* 559.066 *)