Clever Geek Handbook

An intuitive explanation of the integral. Part I - from multiplying natural numbers to Newton and Leibniz

0. Foreword

Mathematics is a versatile, powerful and elegant branch of knowledge. In essence, its subject and meaning cannot be shared with the most fundamental sections of philosophy - logic, ontology and the theory of knowledge. That is why it concerns directly or indirectly all aspects of any applied or theoretical knowledge.





Unfortunately, it so happened that to many (and to me) it sometimes seems too complicated, inaccessible, science for the elite. Meanwhile, it only seems so! Of course, it requires intellectual tension, memory, imagination and much more, like many other intellectual pursuits.





Its distinctive features are:





  1. use of a special sign system (numbers, letters of different alphabets, language rules, etc.),





  2. logical rigor (concepts, definitions, judgments, inference rules are set in an explicit and precise form),





  3. sequence (you won't understand point 3 if you don't understand points 1 and 2),





  4. high density of information per unit of text (often there is much more sense in the text than in texts of other content).





It is easy to show that any intellectually developed person regularly uses the same mental constructs as mathematics. When we say, let's consider ten any operations (algorithm) like a recipe or a simple program, or consider a particular case of a phenomenon, determine its properties, relationships with other phenomena, study the structure - we resort to universal ways of thinking that are characteristic of any knowledge, including mathematical.





This article would never have come to light if the educational literature were so perfect that it could easily explain what an integral is. After reading dozens of books and articles, I can say with confidence that none of them explains all the nuances of this issue in such a way that everything is absolutely clear to an average, inexperienced person.





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Cumulative \ sum \ approx S (x) \ approx S_1 (\ Delta x_1) + S_2 (\ Delta x_2) +… + S_n (\ Delta x_n) \ quad (ii.1)

a b a = x_0 b = x_n.





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\ lim_ {n \ rightarrow + \ infty} S (x) = S_1 (\ Delta x_1) + S_2 (\ Delta x_2) +… + S_n (\ Delta x_n) \ quad (ii.2)

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v (t) = S '(t) = \ lim_ {t \ rightarrow 0} \ frac {\ Delta S} {\ Delta t} \ quad (iv. 1)

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dif \ quad S (t) \ rightarrow S '(t) = v (t) \ quad (iv. 2) int \ quad S (t) \ leftarrow S '(t) = v (t) \ quad (iv. 3)





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int \ quad F (t) \ leftarrow S '(t) = v (t) \ quad (iv. 4)

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\ Integral \ sum = F (b) - F (a) \ quad (iv. 5)

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[4]. 3,5 · 2 + 3,5 · 0,2 = 3,5 (2 + 0,1) = 3,5 · 2,1





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[6]. - — x , - x_1, x_2, ..., x_n A (x) f (x) . , , f (x) A (x), A (x) = f '(x) F '(x) = f (x).





[7]. That is F (x_1) \ neq F '(x_1). For example, suppose a function is given by an expression F '(x) = 2x + 3. Then, when x_2 = 2, F '(x_2) = 9and value F (x_2) = 18. If F '(x) = 0x + 3. Then, when x_2 = 2, F '(x_2) = 3and value F (x_2) = 6.





[8]. Let there be a point, the number 7 and 10, to find the size of the interval between these values, you need to find the difference, that is, 10 - 7 = 3.







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