I'm not sure when you're dating the Industrial Revolution.
Before the Revolutionary War, it seems that all octagonal rifle barrels had some taper and flare. The degree varied. Seems like in the 1780's and 1790s, this became less pronounced and sometimes tapered barrels were used, with less flare at the muzzle. I've seldom seen plain straight barrels on originals that can be reliably dated before 1790. Swamped barrels are seldom seen on guns apparently built with new barrels after 1820.
As important or more important than "swamped" is mass, and the two are related functionally. It is very hard to get the right architecture on a pre-Revolutionary rifle with a barrel much under 1" thick at the breech. A 7/8" straight octagon on a Revolutionary War era gun does not look as right as a straight 1" barrel. But few want to heft a straight octagon, 1" barrel, .50 caliber, 44" long. The weight is considerable and the balance point is way out front. A swamped barrel of the same breech diameter is lighter and the balance point is closer to the breech. That mass at the breech and the balance point, not the curves, are the best reasons to use a swamped barrel when it fits the style.
Folks who grew up carrying muzzle-light centerfire rifles and shotguns weighing under 8 pounds have a hard time transitioning to 9 and a half pound rifles- and that's what you'd have with a 1", straight octagon, .50 caliber longrifle.