Spivak's Definition of Integrals vs Stewart's

[Simpl0S]

Hello.

I finished working through Spivak's Calculus 3rd edition chapters 13 "Integrals", and 14 "The Fundamental Theorem of Calculus". By that I mean that I read the chapters, actively tried to prove every lemma, theorem and corollary before looking at Spivak's proofs, took notes into my notebook, and memorized the definitions, lemmas, theorems and corollary. After that I went on to solve some problems from chapter 13. I supplement exercises from Spivak's Calculus with Stewart's Calculus 7th edition because I find that Spivak lacks in mechanistic/computational exercises and Stewart includes mainly types of those exercises.

Up until now everything was working out good but I came to notice that the definition on integrals from Spivak and Stewart differ.

Spivak's Definition:

Stewart's Definition:

Extracting information from this thread, from post #7:

https://www.physicsforums.com/threads/is-pure-math-useless.284355/#post-2031329

I came to notice that Spivak's Definition is called the darboux definition (Correct me if I am mistaken). I kinda hate this part of Spivak that he is trying to avoid naming the theorems, properties, definitions... It makes it hard when trying to reference them.

Which one is the more accurate or more used or more practical definition? What are the strengths/weaknesses? I prefer Spivak's definition because it appeals to me more intuitively after going through his inf's and sup's chapters and Spivak's definition is, to me, more beautiful.

I also have simply missed how to integrate. I think I have not been able to extract the big picture of integrals and the connection with the fundamental theorem(s) of calculus. Must I always use Spivak's definition with lower sums and upper sums when I want to integrate? How do I integrate? I had calculus in high school (it was called Analysis and it was non-rigorous, e.g. the teacher said: "Memorize the derivative rules and use them to find the derivative of the following function, then find maxima, minima, point of inflection"); in high school I remember that we used integrals in terms of what is called in German "Stammfunktion" which was the function F(x) = integral dx, and we used it to find the area under a curve and to find the area of a body which rotates across the horizontal axis or vertical axis. And again there we had some integration rules given by the teacher and we just used it in a computational way.

But now after reading Spivak I have no clue on how to integrate, for some reason. I might have missed or not completely understood the point of the chapters. Any help?

 

[fresh_42]

Maybe you want to read this insight: https://www.physicsforums.com/insights/omissions-mathematics-education-gauge-integration/
However, it only contains a reference for the equivalence of Darboux and Riemann integral.

 

[mathman]

Spivak's looks like Darboux, while Stewart's looks like Riemann. These are equivalent.

 

[Simpl0S]

@fresh_42 Thank you for the article. It was an enlightening read into my question and it was nice to see what comes after the Riemann integrals.

Spivak also has an Appendix called "Riemann Sums" where he points out what a Riemann sum is. But even there Spivak does not use limits in his definition of a Riemann sum, whereas Stewart defines it with a limit. But I think I am convinced that these are equivalent.

Does anyone have an answer to my other questions on how do I intergrate and what the connection is to the fundamental theorem of calculus? Or are these two chapters just theoretical introductions to the notion of integrals and later on Spivak presents chapter on "How to intergrate"?

 

[fresh_42]

@fresh_42 Thank you for the article. It was an enlightening read into my question and it was nice to see what comes after the Riemann integrals.

Spivak also has an Appendix called "Riemann Sums" where he points out what a Riemann sum is. But even there Spivak does not use limits in his definition of a Riemann sum, whereas Stewart defines it with a limit. But I think I am convinced that these are equivalent.

Does anyone have an answer to my other questions on how do I intergrate and what the connection is to the fundamental theorem of calculus? Or are these two chapters just theoretical introductions to the notion of integrals and later on Spivak presents chapter on "How to intergrate"?

Spivak's limits are disguised in the supremum and infimum over all possible partitions.

Stammfunktion $F(x)$ of $f(x)$ is in English antiderivative, because $\frac{d}{dx}F(x) = f(x)$, i.e. "given $f$ searching $F$" is the opposite of a differentiation, which is from $F$ to $f$. The connection to the fundamental theorem of calculus is, that $\int_a^b f(x)dx = F(b)-F(a)$. The underlying question is: Which function $F$ must I find such that $F'(x)=f(x)$ my given function $f$.

How to integrate? How do you differentiate? You gather a lot of rules and formulas which help you to integrate, and these formulas are proven by the application of the definitions above. It is often tricky to find the right way of integration and even more often only possible by numeric (algorithmic) methods, if the function to integrate isn't in a list of integrals or cannot be attacked by one of the formulas.

 

[Simpl0S]

I think I expressed myself wrong. Let me explain my problem.

Spivak goes on and develops the intuition of the integral and then defines it as mentioned in post #1. After that he goes on and presents Theorem 2, which is a re-statement of the definition of integrals, because according to him, it is often difficult to work with inf's and sup's. He justifies this by integrating the function ƒ(x) = x and ƒ(x) = x^2 directly using the (Darboux) definition of integrals. Then to the reader it will be clear that this approach is quite tedious and then he states that the next chapter (chapter 14 The Fundamental Theorem of Calculus) will present a simpler way of integration. Then he presents more theorems, like "if ƒ is continuous on a closed interval, then ƒ is integrable on the closed interval", or "if two functions are integrable on a closed interval, then the sum of those two functions is integrable on the closed interval", &c. Now in Chapter 14 he presents the first fundamental theorem of calculus and the second. And now here is my problem. I, for some reason, was unable to extract how does one use the fundamental theorem of calculus to integrate? Can you show me an example by integrating a function? The only method I learned so far from Spivak is by applying directly the definition of integrals.

 

[lurflurf]

$$\int_a^b\,f (x)\mathrm {d}x=F (b)-F (a)\\
F^\prime (x)=f (x)$$
In other words find an antiderivative evaluate it at the endpoints and subtract.

 

[Simpl0S]

Thank you all for the replies and help. It clarified everything. I also skimmed forward to Spivak's chapter 19 "Integration in elementary terms" and there he presents what other books would label as "Integration techniques". So for now I have to deal with using the definition and the Fundamental Theorem(s) of Calculus to integrate and get through the chapters with trigonometric functions ans exponential and logarithmic.

I have one more question. Are the trigonometric functions, exponential and logarithmic functions called transcendental functions? If yes why so and what does this terminology mean?

 

[fresh_42]

I have one more question. Are the trigonometric functions, exponential and logarithmic functions called transcendental functions? If yes why so and what does this terminology mean?

As far as I can see it, a transcendental function is a function, which is not algebraic. And a function $f(x_1, \ldots , x_n)$ is algebraic, if it solves an equation $p(f(x_1, \ldots , x_n),x_1, \ldots , x_n)=0$ for a polynomial $p$.

So the question is: Can there be such a polynomial for the functions you listed?

 

[Simpl0S]

As far as I can see it, a transcendental function is a function, which is not algebraic. And a function $f(x_1, \ldots , x_n)$ is algebraic, if it solves an equation $p(f(x_1, \ldots , x_n),x_1, \ldots , x_n)=0$ for a polynomial $p$.

So the question is: Can there be such a polynomial for the functions you listed?

Put in those therms then I would say no, for the functions I have listed there is no such polynomial, thus making the trigonometric and exponential function (and their inverses), transcendentals.

Thank you.