Surface area of a paraboloid

i have to evaluate surface area.. the surface looks like a quadraticsurface on x-y plane . i broke the whole curve in 3 parts and performed curve fitting on it. after this the part of the curve in positive x direction is revolved around the y axis creating a paraboloid type of surface(Note: the axis of fitted parabola is diff from the y axis so the curve will differ in positive x direction from negative x direction)

now this is how i m trying to evaluate the surface area of the surface i generated.

this is d code i was using to findout the surface integral
syms r theta
x=2.515251898 + r*cos(theta) ;
z=2.515251898 + r*sin(theta) ;
y= (x.^2 + z.^2 )*0.0152151093240165
R=[x y z]
Ru = diff(R,r)
Rv = diff(R,theta)
g = cross(Ru,Rv)
rf = 30.0499755714919
t= g(1)^2 +g(2)^2 +g(3)^2
S = int(int(sqrt(t), r,ro,rf), theta,0,pi());

the problem is it is not able to evaluate the surface integral.. it's giving errors
Warning: Explicit integral could not be found.
is their any other way to evaluate the surface area or if there is any problem in coding please help me out
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