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.