I assume some basic knowledge on differentiation and basic algebra, while some good basic knowledge on real analysis like technical refining on Riemann sum, MVT (mean value theorem) or upper/lower limit of Riemann sum, are not assumed but maybe discussed a bit here. The issue discussed will be less vigour then undergrad. course.
We start with FToC (fundamental theorem of calculus), anti-derivatives, then some technical work on substituion and by part; some real application on area and volume. That should be enough to cover most A-level techniques. The application goes deeper for differential equations, some simple contour integration, integration on polar coordinates and multiple integral. To end with we will focus on the art of integration, some special and beautiful techniques which kills many ugly integrands. Questions will be from A-level materials and the series "Today's calculation of Integral" (TCoI).
Infinitesimal
Strictly speaking, infinitesimal is a series
However, this definition is quite academic and we may want to make it easier. A easy interpretation is that, infinitesimal is a quantity that "infinitely close to zero but non-zero". If infinitely large on a real number line is well-defined, we can say infinitesimal is the reciprocal of infinitely large numbers.
In differentiation, we have learnt the first principle on differentiation, showing that x-change in "infintiely small interval" dividing y-change in "corresponding interval" equals to the instantaneous slope. The term
The relationship between infinitesimal and integration is as follows.
Notation. In the following sections,
Riemann sum as the definition of integration
In this level we are not going to discuss different definition of integration (unlike differentiation, integration has many different valid definition) like Lebesgue integration because that's far too difficult. Instead we turn it into a reality concept.
Imformal definition.
Definition. (Riemann integral) For a function f, define
Fundamental theorem of calculus (FToC)
FToC I: Let f be function in close interval [a,b], define
FToC II: Let F'(x) = f(x). Then
The proof of FoTC I used MVT (mean value theorem) in the vigorous proof, and here is an easier proof (which should be avoided in higher level):
Where
We can assume that the first integrands has only one extra term because it only has a difference of
If an appropiate
Therefore we have
We shall prove FToC II by the concept of infinitesimal (instead of limit language in real analysis) because the idea is same as definite integral later.
Let
Here is an analogy to make the above statement easier to understand. Let f(x) = 2x. It's clear that f(3) - f(1) = 4. The "y-change of f from 1 to 3 is +4". Now we make things smaller:
"y-change of f from 1 to 1.5 is +1", similarly, "y-change of f from 1.5 to 2 is +1" and so on. If we add them up we can still get the same result.
Now
Warning. I shall reaffrim here that the above proof is informal, but the idea is the same.
Indefinite integral and uniqueness
Definition. Indefinite integral
We now show that integrand of each function is unique except a different constant at the back.
Assume
Since integration is the inverse of differentiation, we have
Significance:
1) As stated, integration and differentation is the inverse each other. i.e.,
2) That is, if we can find F so that F' = f, then integrate f gives F. We can solve many simple integrands by such "experiences".
We have now clerify the technical problems that we need in our current level so that we can head into some practical work next time. For those who is not capable to realize the technical problems, especially FoTC, should try to memorize the fact that integration is the inverse of differentiation, and come back after some basic real analysis knowledge.
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