That is a very cool machine. It appears to me that the machine is doing a Simpson's Rule integration by measuring the lengths of numerous fixed width strips, then adding the areas together.

As a civil engineer, I have used planimeters for years. A polar planimeter integrates polar coordinates. It's roughly similar to using an infintesimal application of the double meridian distance method of finding areas, except using polar coordinates instead of rectangular coordinates. http://en.wikipedia.org/wiki/Planimeter

You should see this one!

http://www.youtube.com/watch?v=s1i-dnAH9Y4

It's 41 minutes long and entitled "Mechanical Computer - Basic Mechanisms In Fire Control", and it's a 1960's military educational movie about mechanical computers aboard ships for fire control.

They explain how everything happens analogue with mechanical parts: addition, multiplication, 2D functions, etc...

I found it extremely interesting.

Absolutely fascinating. I now have a much better understanding of mechanisms I have seen over the years including this one (photos). In high school I worked summers for a surplus dealer who gave me all the leftovers from an aircraft navigational computer for salvaging all the transmitters, receivers and resolvers which he could resell. I've treasured all the pieces (many gears, bearings, differentials, etc.) and have wished I could find one of these intact someday just to display in a big bell jar or something. Incredible design and engineering!

That was friggin' awesome. 1st time I've ever seen one of those. If I ever thought about it I would think they just estimated. Thanks for sharing!

Curious - but it is performing a numerical approximation - not calculus.

TomC