Clear[R, P, x, y, 
        z];
u = 
        Solve[{x^2 + y^2 == P^2,  z + y == P,  
            x^2 == y*(2*R - y)}, {x, y, z}];

Simplify[%]

Vr = π/3*y^2*(3*R - y)  /.  
        u[[2]]
Vp = π/3*z^2*(3*P - z)  /.  u[[2]]

Expand[Vp]
Expand[Vr]

Vol = Simplify[Vp + Vr]

dp = 
        3/(4*π*P^3); 
dr = 3/(4*π*R^3);
Vl  =  
      Vol*Sqrt[dp*dr]*R/(y*(R - y))  /.  u[[2]]

Simplify[%]

plot1 = 
      Graphics[{Circle[{2.5, 0}, 2.5], 
          Circle[{0, -2.5}, 2.5], {Dashing[{0.03, 0.03}], 
            Circle[{-2.5, 0}, 2.5], 
            Circle[{0, 2.5}, 2.5]}, {Thickness[0.01], Circle[{0, 0}, 1], 
            Line[{{0.2 , 0.9798}, {0.2, -0.9798}}] , 
            Line[{{0.9798, -0.2 }, {-0.9798, -0.2}}]}}]

Show[
      plot1, {AspectRatio  ->  Automatic, PlotRange  ->  All, 
        Axes  ->  True, Frame  ->  True}]