Question

You work for a surveying company and your manager wishes to find the length of a river.

As you examine the river, you notice that a particular oddity manifests. From a very general view, the river looks like a curve. However, as you zoom in and take a closer look, you see that the river has ridges and nooks. As you magnify further you see that the ridges and nooks have smaller ridges and nooks of their own.

Describe an idea or method that would find the closest approximation of the length of the river.

Answer 1

I will designate points along the bank of the river. Each point will be a distance - say 1 meter away from the next and along the bank of the river. I will then count the total number of points I had to lay along the bank of the river

If there are n points then the length of the river will be estimated as (n-1) meters (as distance between 2 points is 1 meter)

The 1 meter was chosen because it is a quick measurable quantity, and because it is not so large that the actual shape of the river will be lost in placing several points 1 meter apart.

This method is similar to approximating the length of a curve by a definite integral.