My dad designed Frequency Response Analyzers in the 60's and 70's that used OpAmps to compute Fourier Integrals and then "solve the right triangle". That is, relative to a sine wave it generated, it would take a return analog signal, multiply by sine and cosine waves, integrate both, and then take the square-root of the sum of the squares and also the arctangent of their ratio, to finally output amplitude and phase-shift of the return signal. All as analog voltages; all with OpAmps (and a tiny bit of TTL to turn the integrators on and off). It simply wasn't feasible to do this digitally; it's not just that doing a real-time FFT in the digital sphere was cost-prohibitive at the time, but also the A-to-D converters weren't fast enough to keep up (at the required accuracy).

Typical users went from aerospace and automotive ("will our wing or chassis oscillate badly when we hit a thermal or pothole"?) to audio (that's how they would make those Frequency Response charts for speakers) to hard disk drive head servos ("if we send the head from track X to track Y, will it get there directly, or "ring" -- slightly overshoot and then overcorrect for a while?").

One fun detail: eventually they had to encase their OpAmps in special little heaters that would keep them at a constant temperature, to avoid drift as the ambient temperature changed.

You can still occasionally buy one surplus on eBay for a few hundred dollars; they were about $10k new (in 70's dollars).