Find the region between y=e^x and y=e^-x rotated around the y axis. The bound is x=1. Would I use shell method for this or washer?
Shells would work. I don't see how "washers" would be used here.
You could also do this is two separate parts. When x= 1, y= is a bound and at x=-1, is a bound. That is, for y= 0 to the right and left bounds of the area are the straight lines x= 1 and x= -1. The figure is a cylinder, having area . You can do the volume from up to y= 1 (where the two graphs intersect) by discs.