My experience was the same as Orgopete's. The first NMR I used on a hands-on basis, back in 1984, was a Varian A60. These were scanning analog machines - the radio frequency signal projected through the sample would gradually scan from 10ppm to 0ppm (or whatever boundaries you set) and a chart printer indicated the absorption as the machine scanned the sample. To get better resolution, you just scanned slower. When you had scanned the sample once in a frequency readout mode, you went back and scanned it again in an integral mode. Essentially the same, but it didn't return to zero, it just kept accumulating signal. You would manually zero it after each set of peaks. I heard at the time that previous instruments lacked an integral mode and that you could Xerox the spectrum, zooming in on your peaks (there was one Xerox machine in the building that actually had a zoom function!) and then cut them out and weigh them to get the integral, but I never had to do that for an NMR.
Also, like Orgopete, I did have to do that for GC and later for LC traces which also had simple paper traces. It took much longer to get peak integration for those systems, and I never had access to a digital integrator. For simple traces where the peaks were reasonably triangular, however, it was faster and sufficiently accurate to use the height of the peak and width at half-height to calculate the peak area.