Could anyone help me propose the surface integral, to solve this problem extracted from the book "Calculus of vector functions, page 369"?
Find the total mass of a spherical film having density at each point equal to the linear distance of the point from a single fixed point on the sphere.
Since the point is arbitrary, you might as well make it the south pole: $$\rho(x,y,z) = \sqrt{x^2+y^2+(z+r)^2}.$$
Now use the fact that $x^2+y^2+z^2=r^2$ and get $$\rho(x,y,z) = \sqrt{2r^2 + 2 r z}.$$
Then you are trying to calculate $$\int_{x^2+y^2+z^2=r^2} \rho(x,y,z)\,dA.$$ Notice that $\rho$ depends only on $z$, which might suggest a particularly useful coordinate system to choose when performing this calculation. Can you take it from here?