
Title: Integral Symbol

Series: Advent of Mathematical Symbols

YouTubeTitle: Advent of Mathematical Symbols  Part 31  Integral Symbol

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Subtitle in English
1 00:00:00,443 –> 00:00:03,744 Hello and welcome back to the next mathematical symbol.
2 00:00:03,786 –> 00:00:07,408 Which is the integral symbol denoted with this long S.
3 00:00:07,608 –> 00:00:12,928 So whenever you see an integral, it always uses this stylized S
4 00:00:13,257 –> 00:00:19,725 and there you might already know, originally it comes from the word “sum” or the latin word “summa”.
5 00:00:20,157 –> 00:00:29,261 Indeed if you want to see the integral as the area between the graph of the function and the xaxis, a sum helps us to calculate it.
6 00:00:29,571 –> 00:00:36,555 I would always say, if you start calculating integrals, you first start calculating the area of rectangles
7 00:00:36,943 –> 00:00:43,193 or more precisely, you would approximate the actual area with the sum of simple areas
8 00:00:43,829 –> 00:00:47,757 and usually this approach is known as the Riemann integral.
9 00:00:48,043 –> 00:00:52,568 So we would write: we have a sum over areas of rectangles.
10 00:00:52,943 –> 00:00:58,332 Now, one rectangle has a height of f(x) and the width of delta x.
11 00:00:58,786 –> 00:01:05,692 So you see, this is not complicated at all. We just read the value of the function and the distance on the xaxis
12 00:01:06,100 –> 00:01:09,647 and then summing it up gives us the whole area
13 00:01:09,847 –> 00:01:15,953 and then you should see, in order to make the approximation better, we make the rectangles smaller and smaller.
14 00:01:16,214 –> 00:01:21,018 So in some sense we have a limit process, where delta x goes to 0
15 00:01:21,218 –> 00:01:28,711 and then the result of this limit process is the actual area. Which we now don’t denote with this sigma, but with this long S
16 00:01:29,229 –> 00:01:33,230 and moreover the delta x is then written as a dx.
17 00:01:33,430 –> 00:01:39,538 So this would explain why we have exactly this nice notation here for denoting an integral.
18 00:01:39,738 –> 00:01:47,066 Moreover I can tell you. Besides this Riemann integral here, there are also other approaches to define integrals
19 00:01:47,266 –> 00:01:53,725 and indeed they change the notation a little bit, but they don’t change the long S for denoting the integral.
20 00:01:54,214 –> 00:01:58,986 Now, if you want to learn more about this, I have a lot of videos about the Riemann integral
21 00:01:59,043 –> 00:02:02,229 and about the modern version, the Lebesgue integral.
22 00:02:02,800 –> 00:02:08,118 therefore I would say let’s meet there or in the next video about mathematical symbols.
23 00:02:08,318 –> 00:02:10,286 Have a nice day and Bye!