1. 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 write-up that must be handed in. Sorry if this is jumbled/empty rambling.

2. 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.

3. 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

4. When you switch the order of integration it becomes positive