Some help understanding integrals and calculus in general

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Discussion Overview

The discussion revolves around understanding the motivation behind the concept of definite integrals in calculus, particularly in relation to the fundamental theorem of calculus. Participants explore historical perspectives, intuitive notions, and the definitions of integrals and derivatives, with a focus on their applications and implications in mathematics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the practicality of computing the area under a curve and seeks to understand the motivation behind the definite integral, suggesting that Newton could not have known about the fundamental theorem beforehand.
  • Another participant proposes that Newton likely had an intuitive understanding of the relationship between integration and differentiation, even if he could not formally prove it.
  • Several participants mention applications of integrals in solving differential equations, indicating that integration is essential in various real-world contexts.
  • One participant references Archimedes' early work on calculating volumes as a precursor to the fundamental theorem of calculus.
  • Another participant discusses the concept of "anti-differencing" in the context of finite differences, suggesting it provides an intuitive understanding of the fundamental theorem.
  • Concerns are raised about the definition of the definite integral, with some participants arguing that it should be understood as a limit rather than solely as an area under the curve.
  • There is a discussion about the definition of a function in terms of an integral, with participants exploring how this relates to the connection between derivatives and integrals.
  • One participant emphasizes that the definite integral has many applications beyond representing area, challenging the notion that it should be viewed only in that context.

Areas of Agreement / Disagreement

Participants express various viewpoints about the motivation and definitions surrounding definite integrals and the fundamental theorem of calculus. There is no consensus on the historical understanding of these concepts or the best way to define integrals, indicating that multiple competing views remain.

Contextual Notes

Some participants express uncertainty about the historical context of Newton's discoveries and the definitions of integrals, highlighting the complexity of these mathematical concepts and their interpretations.

  • #31
Sho Kano said:
So what ##\int _{ a }^{ x }{ f(t)dt } ## is is an indefinite integral?
Yes.

So you're saying it's an indefinite integral, not a definite integral.

Considering ##x## to be variable, its an indefinite integral, not a definite integral.

Mathematical terminology is not completely consistent. As illustrated by the Wikipedia article on "Antiderivative", some people use the term "indefinite integral" merely as a synonym for "antiderivative". If we want to understand the content of the FTC, we have to use a definition of "indefinite integral" that is different from the definition of "antiderivative".
 
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  • #32
Stephen Tashi said:
Yes.
Considering ##x## to be variable, its an indefinite integral, not a definite integral.

Mathematical terminology is not completely consistent. As illustrated by the Wikipedia article on "Antiderivative", some people use the term "indefinite integral" merely as a synonym for "antiderivative". If we want to understand the content of the FTC, we have to use a definition of "indefinite integral" that is different from the definition of "antiderivative".
Stephen, I have edited my earlier post (post 30).
 
  • #33
Sho Kano said:
That would mean a definite integral has numbers as the upper/lower limits, and an indefinite integral is a function.
Yes.
And so what that ultimately means is that the FTC (part I) proves that indefinite integrals can be differentiated to get the integrand. SO, that means that indefinite integrals are a type of anti-derivative. It is not the situation of 1+1+1-1 vs 1+1 because indefinite integrals were not "thought of" as an anti-derivative before.
Yes.
 
  • #34
So from this, it appears that part I of the FTC establishes a connection between anti-differentiation and indefinite integration. But what this also means is that you can take the derivative of an indefinite integral, and end up with the original function - meaning differentiation "undoes" indefinite integration. This is why people say the FTC establishes a connection between differentiation and anti-differentiation. Correct?

If this is what the FTC does, then what does the bit on Wikipedia about all continuous functions having integrals?
 
  • #35
Sho Kano said:
So from this, it appears that part I of the FTC establishes a connection between anti-differentiation and indefinite integration. But what this also means is that you can take the derivative of an indefinite integral, and end up with the original function - meaning differentiation "undoes" indefinite integration. This is why people say "the FTC establishes a connection between differentiation and anti-differentiation. Correct?

Yes. Keep in mind the FTC assumes the original function must satisfy certain conditions, but you grasp the basic idea.
 
  • #36
Stephen Tashi said:
Yes. Keep in mind the FTC assumes the original function must satisfy certain conditions, but you grasp the basic idea.
I still have one more thing I'm uncomfortable with (last paragraph of post 34): If this is essentially what the FTC does, then what does the bit on Wikipedia about all continuous functions having integrals? I mean, in the proof, we are assuming a continuous function and then taking the indefinite integral of it.

...and then differentiating it, which implies F is a differentiable function, meaning that f has to be continuous? Is this why Wikipedia says what it says?
 
  • #37
Suppose g is continuous and f = ∫g. Then you also know that df/dx = g. But there are other functions, h, where dh/dx = g almost everywhere and h ≠ ∫g. An example is h = f+c, where c is the Cantor function. That is why articles like Wikipedia have to state things carefully. So for any continuous function, g, there is guaranteed to be one function, f, which is the anti-derivative of g, but there may be other functions like h that complicate things.
 
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  • #38
FactChecker said:
Suppose g is continuous and f = ∫g. Then you also know that df/dx = g. But there are other functions, h, where dh/dx = g almost everywhere and h ≠ ∫g. An example is h = f+c, where c is the Cantor function. That is why articles like Wikipedia have to state things carefully. So for any continuous function, g, there is guaranteed to be one function, f, which is the anti-derivative of g, but there may be other functions like h that complicate things.
So what I have from this is that it guarantees an anti-derivative, but not all, such as special cases as the cantor function.
 
  • #39
Sho Kano said:
Is this why Wikipedia says what it says?

I'm not sure exactly which passage in the Wikipedia you are thinking about - and I haven't watched the videos of the proof of the FTC. Which proof are you talking about? - the videos or the Wikipedia's ?
 
  • #40
  • #41
Sho Kano said:
I meant the section on Wikipedia saying "This part of the theorem guarantees the existence of antiderivatives for continuous functions.[2]"
But you spoke of continuous functions "having integrals", not about them having antiderivatives.

what the FTC does, then what does the bit on Wikipedia about all continuous functions having integrals?
 
  • #42
Stephen Tashi said:
But you spoke of continuous functions "having integrals", not about them having antiderivatives.
This really ties back to what FactChecker said right? I'm thinking anti-derivatives are a whole set of things, while indefinite integrals are one specific set, or a kind of anti-derivative (kinda like a special case). That's why Wikipedia has to be careful with what they write down.
 
  • #43
Interesting you should suggest Newton might not care about the area under a curve. The area "under a curve" in Cartesian coordinates is similar to the area with in a region between two values of theta bounded by the corresponding radius vectors ( in polar coordinates). Newton was super-interested in this area.

Newton realized Kepler's 2nd law, which states the orbit of the planets sweep out equal areas in equal time is a consequence of the conservation of angular momentum. Newton showed the ellipse (see Kepler's first law) follows from Newton's law and an inverse square gravity source at the ellipse focus. Suffice it to say Newton would have been very interested in areal regions between curves.

Newton was pretty smart.
 
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