Minding the Gap

Here are two ways to draw a disc:

• Make a circle and fill it in
• Draw a bunch of rings with a thick marker

The amount of “space” (area) should be the same in each case, right? And how much space does a ring use?

Well, the very largest ring has radius “r” and a circumference 2 * pi * r. As the rings get smaller their circumference shrinks, but it keeps the pattern of 2 * pi * current radius. The final ring is more like a pinpoint, with no circumference at all.

Now here’s where things get funky. Let’s unroll those rings and line them up. What happens?

• We get a bunch of lines, making a jagged triangle. But if we take thinner rings, that triangle becomes less jagged (more on this in future articles).
• One side has the smallest ring (0) and the other side has the largest ring (2 * pi * r)
• We have rings going from radius 0 to up to “r”. For each possible radius (0 to r), we just place the unrolled ring at that location.
• The total area of the “ring triangle” = 1/2 base * height = 1/2 * r * (2 * pi * r) = pi * r², which is the formula for area!

Yowza! The combined area of the rings = the area of the triangle = area of circle!

This was a quick example, but did you catch the key idea? We took a disc, split it up, and put the segments together in a different way. Calculus showed us that a disc and ring are intimately related: a disc is really just a bunch of rings.

http://betterexplained.com/articles/a-gentle-introduction-to-learning-calculus/