cooltee13 said:
-This is what I've figured out so far:
The area of the curve under the graph can be approximated using calculus, hence its finite. And if I revolve it, the volume is finite, because the area used was finite. However To get the surface area I would need to rotate the length of the curve, which is inifinite, so the SA is infinite as well.
I know this, but I just can't figure out a way to prove it mathematically
The area under 1/x is
not finite if you are going out to infinity: you would integrate 1/x from, say, 1 to infinity, which leads to evaluating
(lim x->inf. ln x) - (ln 1) .
The volume is obtained in this way. Imagine an infinitesimal "slice" of area with height 1/x and width dx. When you revolve it around the x-axis, you now have a disk of area
(pi)·[(1/x)^2] and thickness dx. Upon integrating these infinitesimal volumes from 1 to infinity, you would evaluate
(pi)·{ (lim x->inf. [-1/x] ) - ( -1/1 ) } ,
which is finite.
The surface area of this horn is found by integrating the revolved infinitesimal bits of arclength along the curve, which are infinitesimally-wide bands having width dx and circumference
(2·pi) · [ 1 + (dy/dx)^2 ]^(1/2) .
So you would integrate (2·pi) · [ 1 + {1/(x^2)} ]^(1/2) ] dx from 1 to infinity,
which turns out not to be finite.
EDIT: Your statement about the arclength of the curve is essentially correct. The surface area of the curve is found by integrating a term larger than 1 from 1 to infinity. Since the integral of (1 dx) from x = 1 to x = infinity won't converge, integrating a term
larger than 1 certainly won't converge either. So the surface area of such "power-law" horns is infinite, but the volume for a horn using a power-law 1/(x^p) with p > (1/2) will have finite volume.
When I was first learning about integrating "solids of revolution", we were shown a short cartoon involving someone who'd been set the task of painting a house which was built as half of the "horn" produced by 1/x . They learn that they can't do it because, while the volume of the house is finite, the surface of the house isn't. That still amazes me (as a physicist) in that you could
fill the house with a finite amount of paint, but you can't
coat it. (Such are the nature of mathematical infinities.)