integration by scissors

A term coined by a friend to describe an odd method of integration.

Draw the function to be integrated (Within the range to be integrated) on heavy paper. As big as possible. Cut it out and weight it.
This weight, combined with the knowledge of the density of the paper gives you the intergral. Yay.

This is really just a variation on the counting squares method, but it allows for more complicated (ie, curvy) functions. I suppose it wouldn't work on functions too complecated to draw...

In the 1680s the British government needed the value for the total land area of Britain. Although prior surveys had produced adequate maps of the island, none of these maps had any information on area, and a new, costly, elaborate survey was proposed: teams of surveyors would trace the coastline of the entire island, and their readings would be combined to produce, after extensive calculation, a value for the total area.

The astronomer Edmond Halley approached the problem as follows:

1. He traced a map of Britain onto thick cardstock.
2. He cut out the map's tracing.
3. From the remaining cardstock he cut out circles of radius 2 degrees (latitude). These had an easily calculated area.
4. He weighed the cardstock Britain against the circles on a scale.
5. The ratio of weight gave, instantly, the ratio of area.

Thus, Edmond Halley solved a difficult and expensive problem for the Queen and, simultaneously, invented integration by scissors as we know it today.

Notes:

• Halley used this method to calculate the area of a specified shape, as opposed to the area defined by a specified mathematical function, but the concepts are essentially identical.
• If, as is most likely, Halley worked from a Mercator projection map, the value obtained is inaccurate, as Mercator maps exaggerate areas increasingly with distance from the equator. However, Britain is probably limited enough in latitude for his approach to provide a reasonable estimate.