Evaluate the surface integral over a torus

Question

Let $F(x) = \frac{x}{||x^3||}$ and let $B$ be the 3D region formed when the circle $(z - 2)^2+y^2 = 1$, in the yz plane, is rotated about the y-axis. Evaluate $\int_{dB}F \cdot dS$.
This shape $dB$ should be a torus, with its 2D projection on the yz plane as a circle centered at (0,0,2) and a radius of 1. But I am not sure about my parametrization of the torus(I parametrized the inner "empty region" and the outermost circular region and then found their difference): $\Phi_1(u,v) = (\cos v, u, \sin v)$ is the inner circle and $\Phi_2(u,v) = (3\cos v, u, 3\sin v)$ and I used $\Phi_2 - \Phi_1$ as my parametrization. Can someone check this for me or give a hint?