Area of ​a parametrically defined function

I can’t solve the problem; I need to find the area of ​​a parametrically defined function. According to the graph, part of the area has a negative sign.

\[Alpha] = 1.525;\[Beta] = 3.95;\[Chi] = 50;Hr = 58.3;He = 36.6;Bmax = 1.5;ParametricPlot[{\[Alpha] Sinh[Bmax Sin[\[Gamma]] \[Beta]] +   Bmax Sin[\[Gamma]] \[Chi], (Bmax^2 - (Bmax Sin[         ArcTan[Hr/Sqrt[          He^2 + ((\[Alpha] Sinh[Bmax Sin[\[Gamma]] \[Beta]] +              Bmax Sin[\[Gamma]] \[Chi]) )^2]]])^2)^(1/2)     Sin[\[Gamma]] - (Bmax Sin[       ArcTan[Hr/Sqrt[        He^2 + ((\[Alpha] Sinh[Bmax Sin[\[Gamma]] \[Beta]] +            Bmax Sin[\[Gamma]] \[Chi]) )^2]]]) Cos[\[Gamma]]}, {\\[Gamma], -\[Pi], \[Pi]}, AspectRatio -> 1/2]

Here is my solution, which I spotted on the forum:

With[{x = \[Alpha] Sinh[Bmax Sin[\[Gamma]] \[Beta]] +    Bmax Sin[\[Gamma]] \[Chi],   y = (Bmax^2 - (Bmax Sin[          ArcTan[Hr/Sqrt[           He^2 + ((\[Alpha] Sinh[Bmax Sin[\[Gamma]] \[Beta]] +               Bmax Sin[\[Gamma]] \[Chi]) )^2]]])^2)^(1/2)      Sin[\[Gamma]] - (Bmax Sin[        ArcTan[Hr/Sqrt[         He^2 + ((\[Alpha] Sinh[Bmax Sin[\[Gamma]] \[Beta]] +             Bmax Sin[\[Gamma]] \[Chi]) )^2]]]) Cos[\[Gamma]]}, NIntegrate[y D[x, \[Gamma]], {\[Gamma], -\[Pi], \[Pi]}]]

but it does not take into account that part of the area is negative.I will be glad for any help, thanks for your attention.