Archive for the ‘Integral Calculus’ Category

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.

Concentric circles were eventually drawn around the lower bound of the main interval and the area of each annulus was reasoned out through discussion. It was concluded that the upper bound of each sub-interval could be used to represent the length of each annulus, with the span of the corresponding partition equal to its width. It was agreed upon that these annuli, when”sliced” and “unrolled”, formed rectangles whose areas were easily determined.
Lengths, widths, and ultimately areas of at least 3 of these rectangles were recorded and then summed. Some simple factoring revealed a “new” expression with recognizable components. After some simplification, “q” was replaced with “1” and out popped a very good estimate for the area of a circle.
This process was then repeated for f(x)=x^2 and f(x)=x^3 since the factoring skills required for those was also in place. The entire process was then generalized and applied to the function f(x)=x^n and from this, our expression F(x) was 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

Fermat 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.



Courant, Richard., John, Fritz (1999).  Introduction to Calculus and Analytics: Classics of Mathematics. New York, NY:  Springer-Verlag Berlin Heidelberg.

When introducing my high school students to Calculus, I begin by asking them if they would like to see why the volume of a cone is what it is; they invariably respond “yes”.  This leads immediately into some diagrams and discussion on what they already know and information that is missing. I tell them up front that the sum of squares formula will be needed; this is derived through telescopic sums, a rich task in itself.

A diagram of a cone is then sectioned into cylindrical slabs and, through further discussion, it is decided that a reasonable estimate for volume of the entire cone could be determined if these slabs were very thin, there volumes calculated and then summed.  The basic idea behind integration has now been considered and off we go.

By the end of day 2, students have seen how the discrete representation (Riemann sums) gives way to its continuous counterpart (integration) through use of the power rule. We work through several examples with uniform subintervals and then develop the power rule for integration using non-uniform partitions on f(x)=x^n.  The FTC follows shortly thereafter; students are then given some time to practice finding area by setting up simple integrals and working through the process.

Simple differential equations soon follow, as shown below.

Differential Equations.png

Rates of change (and related rates) can very easily be folded into the fabric of our work from thoughtful consideration of the topic shown above.  This is a necessary outcome as the task of deriving integrals of logarithms, trig functions, etc. is too demanding at this level.  Derivatives of these functions are, however, well within the grasp of high school students. Once these are derived, the inverse nature of integrals and derivatives can be exploited to continue on our little journey.

I realize that this approach isn’t for everyone but it works for me and my students seem to enjoy their work as well.

That is all for now. Thanks for reading.



Determine a position “c” in terms of “a” and “b” that will result in triangle ABC having maximum area. Prove that the areas bounded by f(x) and the line segments AC & BC at that position are equal.

Putnam Area

Click on the link provided here to explore the triangle area above.