My goal here is to introduce the ideas of calculus, and methods of summation and integration in the way that can be explained and thought to primary school students. In addition, anyone interested in knowing more about calculus and ideas behind integration can read and, most likely, greatly benefit from this chapter.
There are only a few prerequisites for this integration introduction.
Fig. 1. The Pirate Island at night.
Fig. 2. The Pirate Island during the day.
This chapter will introduce readers with a method to calculate the surface area of an object of an irregular, arbitrary shape.
Students are, for a long time, trained to calculate areas of rectangles, trapezoids, circles, maybe ellipse, and other 2D and 3D objects. They are pushed to memorize formulae in order to pass the tests or math exams. Yet, when they look around, in the real world, there are examples that has nothing to with all that mathematics material the students have been through for many years. Look at the stocks graph, temperature changes, wind direction, car driving, etc. Even kids in 4th grade will have good understanding what is mechanically say, going on in these cases, but, quantification and mathematics are not shown to be, somehow, applicable for those instances yet. One example is that students will be most likely puzzled what would be the formula to calculate a surface area of an object of a completely irregular shape, like the shape of an fried egg, or the shape of an island. This is excellent question, and, moreover, there is no need to keep students away from the answers for years, until they come to secondary school, to get the calculus and pre-calculus classes. Basic ideas are very easy to grasp, and I am sure, this chapter will be an eye opener and new ideas generator for the readers.
Let’s start. You do not need to hunt for a formula in mathematics all the time! Method, approach is way more important than finding, or even worse, memorizing formulae in many cases. And, it's the actual method to calculate a surface area of an arbitrary shape that is the essence of integration, the idea that represents one of the fundamental concepts in calculus.
Let's take our focus from a boring math text, and look around. Where we can apply math? Let's consider the following example. A pirate island. We want to calculate a surface area of a pirate island, let's call it The Pirate Island.
There may be many reason why we need to do that. Maybe we want to see how much of it is covered by forest. Or, we want to make an accurate map for future treasure hunters. Here I want to make one important point. Math can't distinguish between real world and fictional one, like the fairy tale! Check this. Two dragons ate 53kg of coal each. How much coal they ate together? It's up to you to develop reasoning what and why to count, and provide these requirements to math. Math will do the calculations, but can’t distinguish whether the numbers are coming from real or fictional world.
The Island has a completely arbitrary and irregular shape. If we try to search for a formula ready solution, we will not find it, most likely. So, how we calculate the surface area of an arbitrary shape?
To start with, let’s draw the island from the bird’s perspective. Let’s focus only on shape and leave out vegetation, animals, sand, rocks etc. Just the shape of the island in one instance of time. Of course, we have to do that, because the shape of island is constantly changing because of the waves or tides. But, we will assume one shape for our purpose.
Fig. 3. The Pirate Island shape from the bird’s perspective.
Since we know how to calculate the surface of an rectangle, let’s use rectangle to calculate the surface of the island. Let’s cover it with rectangles. We can cover it with triangles too, or with circles, for which we know how to calculate surface area. That will change a little in our approach, and, in addition, rectangles are quite simple too. So, let’s do it. Here is how the Island looks like covered with rectangles.
Fig. 4. The Pirate Island covered with rectangles.
We want to try to cover, as closely as possible, the shape of the island with rectangles, and then compare their surface areas.
If we label each rectangle and note its dimensions we can add them up and obtain the surface area covered by all the rectangles we have used so far. We will immediately notice several things. First is that our covered area is not the actual area of the island, but only close to it. There are errors, there are missed surfaces where the curvatures are. On the good side we can have some idea, from our calculation, after adding rectangles surfaces, what the total area approximately is. It’s a bit bigger than what we got, but we also note that it has a definite limit value, which is actual surface area of the Island. Don’t worry that we are talking here about estimated value. We are doing a good estimate, we have a good approach to estimate. Moreover, the estimation will lead us to determine the exact value of the island surface area.
At this point, let’s see what we can do to get closer to the actual surface area. The first thing that comes to mind is to add more (smaller) rectangles to fill in gaps between present rectangles and the curvature line of the island. That’s correct. Let’s magnify one part of the island and rectangles in it, to see clearly the gaps on surface between rectangle and the curve.
Fig. 5. A magnified portion of the Island.
With the addition of more and more, smaller and smaller rectangles we fill in the gaps denser and denser, and our covering surface area is getting more and more accurate, closer to the real value. However, because of the smoothness of the shore curve, the smoothness of the island shape, there appear to be always gaps between rectangles and the island curve! That is actually very true. Let’s magnify again the portion of the island we have just filled with more rectangles and see how it looks like.
Fig. 6. Again magnified portion of the island with more, smaller, finer rectangles in.
Still more gaps. But, they are smaller too. Hence, the error is smaller. We have a sense, that even if we add more and more rectangles, there will be always gaps, though smaller and smaller, because the curve is smooth and our rectangles have right angles. However, there is one powerful thing that we can notice at this point. While there is a difference between the surface area we are getting by adding smaller and smaller rectangles and the real island surface area, also is the error between the surface area we are getting, and some correct value we are after. Moreover, with this approach, by adding more and more smaller and smaller rectangles we can approach to the correct value as close as we wish. We can get close to any number of decimal places. That sounds like a good achievement, right? If you calculate the surface area of an island that is say 10 square miles with the accuracy of 0.00000000000005 square mile, which is half of the grain of sand area, you seem to have quite a good result. But, you may ask, what is the exact value, and not just approximate, no matter how close to that value, using our rectangles, we get.
There are two answers to this question. One is that, when we add more and more, smaller and smaller rectangles to fill in the gaps, we may be able to see to which value our successive sums are aiming at! We can be 100% certain that there is a value we are after, and sometimes we can actually name that limiting value, and that will be the actual surface area of the Island. Note again, that even we do not “reach” that value with our process of getting rectangles, we still can approach it as close as we wish, and as accurate as we wish, even to millionth of decimal places. When we can say that we can do that kind of approach to some limiting value, the value in question is actual correct value for our way of summation. Moreover, that’s exactly how mathematicians define a limit value of a sum in the process of integration. Integration is nothing more than summation taking into account this limit summation method of our rectangles.
The other answer has to do with the definition of irrational numbers. Irrational means numbers that can not be represented by a ratio of two integers (rational numbers). For instance, square root of 2 or Pi, are some of irrational numbers. Their main property is that they are represented by infinite number of non repeating decimal places. You may think then that these numbers will be infinitely large. But, they are not, because each time smaller and smaller fractions are added, like we added smaller and smaller rectangles in our island and we know that these rectangles are always between island limits and our present calculated surface area. As long as you add smaller and smaller fractions each time, and you know that there is upper limit that can not be exceeded you are dealing with irrational numbers. Try it by yourself. Get any number and start adding more and more decimal places to it! You will see that every time you, actually, can have the upper limit of your number, if you wish, or, you can add more and more decimal places, noting that each time you added obviously a smaller fraction! From the very strict mathematical point of view, what we have just done is the actual definition of an irrational number. They must be defined as limit values as you add smaller and smaller fractions. Sometimes these limit values are clearly visible, sometimes they may not been known, but in each case it has to be proved that they exist and that we can approach them to any accuracy we wish and as close as we wish.
If you still think that, at least theoretically, the coastal island line is still too jagged to be smooth (but continuous!) on any scale, you are right. Another approach is to consider fractal theory, in calculating the coastal line length. Good starting point is Koch snowflake or Mandelbrot works on fractals. That is the subject outside the scope of this post.
Now, back to Island. While we explored our accuracy of calculating surface area, don’t forget what we have started with! We have started even with a thought to try to find ready to use, plug in, formula for the surface of the island. But, that was wrong approach. Then we thought of method of covering rectangles, and by using them we got the correct and very powerful method to calculate, to any accuracy we wish, the island surface area.
We can now generalize our result. Using this method of covering rectangles we can calculate essentially surface area of any shape. Interesting example is calculating surface area of an aircraft tail and rudder section.
Here is illustration. Principle is the same.
Fig. 7. Numerical methods of calculation of a surface area of an aircraft vertical stabilizer and rudder section.
This method of covering rectangles, to call it like that, is the fundamental idea behind calculus and numerical methods in engineering applications. Numerical methods in engineering simply means, in many cases “since there is no ready to use formula, let’s use some approximate method with limiting process for summation, like method of covering rectangles”.
You are now familiar with one more fundamental idea of calculus. Taking the calculation to the limit, in our case we are adding more and more smaller rectangles to get to the limit of the island surface area. Note that! Taking to the limit is what is the basis of the calculus. That’s very powerful thought we came with and powerful method. For instance, if we did not take it to the limit, we may not be able to see what the actual surface area value may be, nor we will be in position to conclude that we will approach that area with any accuracy we want. When you take process to the limit you are already mastering the most fundamental idea in calculus.
Using "covering rectangle" method, method of dividing, then summing the area or line length, or surface areas, (limiting process is involved here, as we have seen) is a common method of solution for arbitrary, irregularly shaped lines, surfaces and volumes. Also, the method is used as well when a function is defined in a closed form, frequently by one or more formulae (where possible) to describe (as a continuous, smooth function) lines, surfaces, and volumes.
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