The connection between integration and area

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Discussion Overview

The discussion revolves around the relationship between definite integration and the concept of area in R², particularly whether calculating area through integration yields the same result as counting unit squares. Participants explore the definitions and implications of integration in this context.

Discussion Character

  • Conceptual clarification, Debate/contested

Main Points Raised

  • Some participants assert that definite integration defines area in R² and volume in R³, questioning the equivalence of integration results and counting unit squares.
  • Others argue that focusing on unit squares reflects an outdated understanding of integration, suggesting that the area of each "piece" in this context is effectively zero.
  • A participant explains that the Riemann integral approximates area by considering rectangles of finite width under a curve, implying a connection to counting squares as the grid size approaches zero.
  • One participant challenges the notion of needing to ensure that integration aligns with intuitive definitions, stating that the integral's definition is explicitly designed to match intuitive ideas of area where applicable.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between integration and counting unit squares, with some emphasizing the need for a deeper understanding of integration beyond simple counting methods. The discussion remains unresolved regarding the adequacy of these perspectives.

Contextual Notes

There are limitations in the assumptions made about integration and area, particularly regarding the definitions and contexts in which they apply. The discussion does not resolve the mathematical nuances involved in these concepts.

Poirot1
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My understanding is that (definite) integration defines area in R^2, volume in R^3 etc, and my question is: would calculating an area in R^2 using integration give the same answer as counting the unit squares. If so, how did they ensure to define integration in such a way as to give the same result as most people's definition?
 
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If you are thinking "Unit Squares", you are not getting the idea of integration. They used to think this way in Ancient Greece - vaguely, if we can just quantize it, we'll find the fundamental unit. This is not good enough. The area of each "piece" really is zero!

We can get most peoples' definition to match up by teaching them THE definition. There are other definitions for integration - to handle discontinuities and various other things.
 
Last edited:
Poirot said:
My understanding is that (definite) integration defines area in R^2, volume in R^3 etc, and my question is: would calculating an area in R^2 using integration give the same answer as counting the unit squares. If so, how did they ensure to define integration in such a way as to give the same result as most people's definition?

The Riemann integral is set up to be the limit if it exists of the area of sets rectangles of finite width that approximate the region under the curve as the widths tend to zero. So it is to a hand-waving approximation the result of counting squares as the size of the grid goes to zero.

It is silly to ask how did they ensure this result, the definition of the integral is set up explicity to match the intuitive idea of area in those cases where the intuitive notion makes sense.

CB
 
Thanks for the responses
 

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