So that we can write this in regular polar cylindrical coordinates, I am going to swap "x" and "z" in the problem:
and .
In cylindrical coordinates, that is and so the two surfaces intersect where or . Integrate with from 0 to and, for each , r from 0 to .
The "differential of surface area" for the cone is .
I did it with the Maple_13; нere's the code for building
A := plot3d([[r*cos(t),r*sin(t),r*sqrt(cos(2*t))],[r*cos(t),r*sin(t),-r*sqrt(cos(2*t))]], t=-(1/4)*Pi .. (1/4)*Pi, r=0 .. 1, numpoints=3000, color="LightBlue", style=surface):
B := plottools[rotate](A,0,0,Pi):
C := plot3d([cos(t),sin(t),z], t=0 .. 2*Pi, z=-1 .. 1, color=pink, style=wireframe, transparency=0.75, numpoints=2000):
plots[display](A,B,C, axes=normal, scaling=constrained, lightmodel=light2, view=[-1.4 .. 1.4, -1.4 .. 1.4, -1.4 .. 1.4], orientation=[70,62]);