
surface of revolution
Given the function f(x) = e^x, with x greater than or equal to 0. This finite area of an infinite region should produce a finite volume of revolution, but then an infinite surface of revolution, a la Toricelli's Trumpet. Is this assumption of mine correct? And if so, what would the proof be like?
Also, I understand that the surface area of revolution would be I = 2pi e^x sqrt[1+e^2x] dx, with bounds 0 and infinity. However, I am not sure how to perform such an integration.
Lastly, this is not really a "homework help", it is just an interest that developed while doing a problem involving Toricelli's Trumpet, but I want to add that part in the writeup that must be handed in. Sorry if this is jumbled/empty rambling.

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.

thank you very much, torriceli's trumpet is the same as gabriel's horn which is how this question came about. i very much appreciate your help and time. however, shouldn't it be 2 pi followed by the INT, given that du is e^x? dx

When you switch the order of integration it becomes positive