What I think I need help with is (forgive my Latex ignorance):

. . ._____

∫√1+1/x dx

I have no idea how to integrate this.

In case I set up the integral wrong, the original question calls for evaluating the surface area of y=2√x rotated around the x-axis (on the interval from 4 to 9).

So first derivative (1/√x)

First derivative squared (1/x)

Into the form of (arc length x 2π):

. . . . ._____

2π∫√1+1/x dx (evaluated on aforementioned interval.)