Area and Volumes of revolution using disc method (1) The disk method to determine the volume of revolution uses the volume of a cylinder of width dx 
    a proof of this involves showing cylinders (disks) above the curve and those below the curve 
    converge at the curve. 
(2) why can't the same logic be used to find the area of revolution, outer area of disk (cylinder, not including faces) - Why can you use dx for volumes but not area (area must use arc length, ds) Obviously
ds and dx are not equal, but area of greater cylinder and area of inner cylinder should converge at 
any point on the curve 
 A: Let's bring ourselves down to the 2D case first. One might ask why can I approximate the area under a curve over some small dx by assuming the top is flat and at a fixed height y, when in fact I cannot approximate the length of an arc over some small dx by assuming the same thing. 
Without getting too "mathy" this should be intuitive. Imagine the arc and splitting the area underneath into small rectangles. Each time we split into smaller and smaller rectangles our approximations always get better. This gives us the notion that as we go to infinity our approximation will be exact. However when we imagine doing the same thing with length our approximation always stays the same and is not converging anywhere. (It is always just the length of the portion of the x axis underneath)
This is similar to the notion that if I draw a right triangle I cannot say that c = a + b. Even though you could imagine that I approximate the diagonal with a zig zag of infinitely small "steps". This doesn't work because I am not reducing the error in the approximation in each application of making the steps smaller. So the length of the zig zag is in fact not approaching the length of the diagonal. (It is staying exactly the same). However the area under the zig zag is clearly visually approaching the area under the initial right triangle. 
So we see that arc length does not do well under these naive approximations. We need to do something different (in this case approximate with tiny diagonal lines rather than horizontal). In the same manner think about where the volume comes from when we do the rotation. It comes from the area under the curve so it is intuitive that the same approach of using dx should work. The surface area however results by rotating a piece of arc. So it seems unlikely that this method would suddenly work. 
For a less hand wavy explanation we can do the same reasoning of asking does the approximation get better as I make things smaller and smaller. In the case of volume the answer is yes. In the case of surface area the answer is no (Do the computation of approximating the surface area of a cone with a cylinder that has the average radius of the region. First just with one cylinder with radius r/2. And then with two cylinders with the top one having radius r/4 and the bottom one having radius 3r/4 and so on. The approximation doesn't go anywhere and clearly is not correct from the get go)
I'm sure you can find more technical answers that dive into analysis of the error of the approximation converging to 0 in one case and not in the other, but in my opinion sticking with the intuitive is the way to go. Hope this helps!
