I’ve always liked using the “onion proof” as an easy approach to showing students why the area of a circle is what it is. I also strive to fold in as many previously learned concepts as possible when introducing new ones. Fermat’s non-uniform partitions provide a very rich approach to deriving areas. This approach as it applies to the circle is shown directly below. As you will see, factoring skills that students have learned in Grades 10 and 11 are put to work and out pops the circle’s area; students like this.
The Setup
A line on [0,1] was first drawn, then partitions based on q=0.5 identified; those partitions were then summed and the total approached 1 (as it should). This idea was then connected to the formula for sum of infinite geometric series (which was also quickly derived). Additional iterations with q=0.9 and q=0.99 revealed that the non-uniform partitions would become “more uniform” as “q” approached “1”. This idea was then generalized over the interval [0,r] and the lengths of each sub-interval determined.
Shown below are the circle and f(x)=x^n. The links provided here allow students to adjust the common ratios for each and view the resulting partitions.
Non-Uniform Partition Exploration: Circle
If we were to sketch a function of area “A” with argument “q”, we would find that it has a point of discontinuity at (1, πr^2). We can say that the function has a limit of πr^2 as “q” approaches 1.
Non-Uniform Partition Exploration: f(x)=x^n
Thanks for looking.
Reference
Courant, Richard., John, Fritz (1999). Introduction to Calculus and Analytics: Classics of Mathematics. New York, NY: Springer-Verlag Berlin Heidelberg.
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