If the region R = { (x,y) : x >= 1, 0 <= y <= (1/x) } is rotated about the x-axis, the volume of the resulting solid is finite. Show that the surface area is infinite. ( don't need to compute the integral )

I know the shape is called gabriel's horn. I think i have it right, I have got the surface area to be S = 2pi (integral from a to b) sqrt(1 + f'(x)^2))dx. I'm not sure if this is right though. Can someone go through this and see if I have done it correctly?

TIA!!!!