I was running around the Internet a few days ago and happened across this web page on Meccano hobbyists. It gives a great explanation of how an analog computer works.
The Differential Analyzer solves differential equations by integration. It makes use of one or more wheel and disc integrators (or Kelvin-disc integrators), interconnected by shafts in various ways to suit the problem equations. The process of integration can be illustrated by the simple example of the acceleration of a car. This can be represented for input by a curved graph showing speed varying with time. Say one wanted to find out the distance traveled in a certain time, say five minutes. The period of 5 minutes can be divided into much smaller intervals of say 10 seconds each, and assuming a constant speed over each interval from the graph, a distance traveled for each interval is calculated. The sum of the distances traveled in successive intervals is then the total distance traveled. The smaller the interval taken the more accurate will be the result. This is called Integration, and is the function performed by the integrator in a continuous manner.
Best of all, with a bit of work these can be made out of Meccano (Erector) sets.