The volume from 0 to infinity is Pi/2.

The surface integral can be evaluated by doing a u sub.

Let u=e^(-x) and get:

2Pi*INT[sqrt(1+u^2)]du, u=0..1

You should get Pi(sqrt(2)+ln(1+sqrt(2)))

It is finite as well.

If you want a function that gives finite volume and infinite surface area, try

integrating y=1/x from 1 to infinity.

This is known as Gabriel's horn.

Sorry, LaTex is down.