You know that I thank God for giving me the opportunity to communicate with the masses. I think this talent comes from my struggle with this subject at an early stage. My ability to convey the courage of this subject stems from my conviction that if I can understand this, anyone can. This is the case with calculus. Keep reading until I show you how this item allows us to calculate the exact area of even bizarrely shaped objects.
Now you have all learned from basic geometry how to find a region of such common shapes as a square or rectangle. Heck Some of you may even remember how to find a region of shapes, like a triangle or a trapezoid. But how would you find an area of irregular shape, like an open rectangle, the upper part of which is formed by some tortuous curve? In other words, draw a rectangle. Now remove the line that forms the top. Draw a curved line from the left side all the way to the right so that the shape is now closed in space. Feel free to make a complex line as complex or confusing as you like, until the curve crosses itself. Can you imagine something like that? Well, then, because these are the kinds of forms, whose areas, calculus will give us the utmost precision!
How calculus does it. Well, it all starts with an approximation. As mentioned in the previous article on Calculus, the two main branches of this subject are differential and integral. The branch that deals with areas of irregular forms is an integral, and this name is what we give to the mathematical object that actually calculates the area.
We all know how to calculate the area of a rectangle. Now imagine that we take this large rectangle with a magnificent top and divide it into five sections as follows. We select five points on the basis of a rectangle so that each divides the base into five equal parts. In each of these points, we draw a vertical line from the point at the base to the point that is on the curve of the curve above. From this point on, we form a smaller rectangle, drawing a horizontal line from a point on a curved curve across to the left, so that the width of this vertex coincides with the width of the base. Can you imagine it?
We do this with each of the flat, spaced points at the base of the rectangle, creating a smaller rectangle from the larger one. You may have suggested that by adding areas of smaller rectangles, we can move closer to areas of larger ones. In fact, it is a process that leads to an integral, a remarkable mathematical object that will give us an exact domain.
Since we have a curved line at the top of a larger rectangle, we will form five smaller rectangles, which in some cases lie inside the larger rectangle, and in some cases lie outside the larger rectangle. You really should try to draw it to see what's going on. In any case, we can get closer to the exact area, if we divide the base into smaller and smaller sections. Therefore, instead of five rectangles, we have a hundred or even a thousand. To get the integral, we use an infinite number of rectangles, in which the point of each of them is zero, and the height is only the height from the base of the larger rectangle to the curve above. Wow, what a sip! Feel it a little.
In essence, to get an exact region of irregular shape, we add an infinite number of rectangles whose width is zero. In fact, we take the limit as the width of each rectangle approaches zero and an infinite number of them. From this striking example, we come to the important conclusion that infinity multiplied by zero is some finite number, which in this case is an exact area of irregular shape! Welcome to the world of calculus. Now you see why this stuff is so fascinating?
