Computing the Area of a Circle

My father once teached me this technique a long time ago but i don't know where he saw it; probably in his lost school book.

Imagine a circle (the portion of area enclosed by a circunference). We saw in the last post that the perimeter is just: 2 times pi times the radius. Now cut it in four equal parts (pies) and rearrange them like in the figure below. Each pie has two straight segments who's lenght equals the radius and a round "side" of a magnitude that equals the perimeter divided by four.
Start again from the begining but cut the circle in eight equal parts, for example. Rearrange like in the next figure. Each pie is like the ones before, the only diference is that the round border has a lenght that equals the perimeter divided by eight:We can do this cutting thinner pies, but an even number of it in order to obtain as many equal oriented pies has those in opposite directions.So in the limit of very (infinite) thin pies we get a figure that is a rectangle with two paralel borders who's lenght equals the radius of the original circle and each of the other paralel borders has the lenght of half perimeter. It's easy to guess that the area of this rectangle equals the area of the original circle and so
I ask you to recall what i said about integration in a post. The difference here is that there are no remaining parts left during the process because we are using pies instead of squares or rectangles. In the limit of very small pies, each pie will look more like a very thin rectangle. That's what easies the calculation: we know how to calculate the area of a rectangle and thats enough.

What is pi?

A circunference is the set of points in a plane that are at equal distance (the radius) from a single point - the center. So a circunference is a closed line with a length we call perimeter. We can also define the diameter wich is just the double of the radius. (see picture below)
The interesting property of a circunference is that we only have to know a quantity (a number) to be able to draw one. Give me a radius and i will draw one. Instead you can give only the diameter. It is also possible to make a circunference knowing only the perimeter. So a circunference is a number and a rule - the rule that all points are equidistant of a center.
So it must not seem strange the fact that for all circunferences the relation between the perimeter and the radious is the same. This means that if you pick a circunference and divide its perimeter by its radious you get always the same number. This could look amazing but remember that you only need one quantity to define the figure. If i we have a radius, there can be only one perimeter correspondent to it. Soand we are free to chose the name of this constant. We can call it Potato, Glass, 3 times BlaBla, etc. But let's defineThen we can measure perimeters and radius of several circunferences and compute the value of pi with the precision we want (not that easy!). In fact, there are other ways to do that only with more or less heavy mathematics and computation but i want to keep this simple. pi value is known for a long time (see here a nice history about pi but remeber to see other sources). You probably know it is something like 3.14...
We have now a definition:On the other hand we have -- now that we know the value of PI -- a way to compute the perimeter knowing the radius:
This is a linear relation. A double radius gives a double perimeter; one third of a radius gives one third of the perimeter; and so on...
There are many interesting facts about this important number and it's easy to find information about it.

Bricolage and Integration

how to calculate the area of odd shapes?
consider the shape of figure 1. cut it in pieces. sum the areas of all the pieces to get the area of the initial shape.
this seems to complicate the problem. we can simplify this by cutting the shape with a square net as in figure 2. its easyer because we already know how to compute the area of each square. the only problem is in the borders where we have excedents (gray parts).

then, consider smaller squares (figure 3) and the sum of the excedent parts will be smaller. the thing is: thinner nets gives smaller excedentsimagine one can chose arbitrary small squares. then, you can decrease the excedent to arbitrary small quantity, eventualy zero. thats the limit that gives the exact area of the shape, and its made of so many squares that they are not even countable! so mathematicians created the concept of integral wich i roughly described here. its a somewhat simple concept but some calculations may be tedious or only aproximated by computer algorithms. integrals do not compute only areas but also volumes and are used in many other things non-geometrical. integrals are represented by a symbol that is much like an "S" and this is because of its definition as a Sum of many little things. (remember that the greek uppercase letter Sigma stands for sums too but of a different kind.) For example:

remark: the definition of integral demands no particular kind of cuts we make to the shape. so we don't have to choose squares. the important thing is that in the limit all pieces become arbitrary small (figure 4)

Math Bricolage Again - A Trapezoid

(see previous posts for triangle and paralelogram areas)

More Math Bricolage

A General Paralelogram

by the way... something that it's not that obvious:

A Very Easy One

math bricolage for fun!

Extending the Factorial Notation

My friend Luis Pereira suggested this work based on the definition of factorial and double factorial from the last post. So in the file below i define p'th order factorial:
where is just the integer part of the quotient inside the brackets. I use the relation
to find the extremum values. Further i get the result
and finally show how this can be used with the example:
Get the file: --- PDF --- DVI ---

Factorial and Double Factorial Notation

I often use the double factorial notation. It's not a big deal and it makes some expressions more elegant.

A technique for integrations

Here is an example of a technique useful in some integrations. I apply it to the improper integral



file: --- PDF --- or --- DVI ---

Evaluating the improper integral of the famous gaussian function

I start by showing a curious method to calculate the well known indefinite integral of the gaussian function


and then i make a further aplication to a little more complicated integral:


(also with positive a)

Get the file in the format you prefer: --- PDF --- DVI ---

About the files

I will post the files in two formats: PDF (for Adobe Reader) and DVI (Yap: comes with MikTex)
I write the files in LaTeX first and then convert it with WinEdt extension from MikTex.

Starting Point

My goal is to make a collection of little things about mathematics familiar for most physics students that i found interesting. Some of the issues will be trivial, others are merelly academic and a few could be a little harder. This blog will be like the bunch of papers that grows in the closet because it dont fit in any classified file folder. I have some things i wrote a long time ago so maybe you find some stuff funny for younger minds.
Please mail me the english errors i make so i can learn and correct. Of course, if you find any algebraic incorrection, want any explanation or want to give a suggestion send it by mail...

cdanielabr@netcabo.pt