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:
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.