Incomplete integration explanation

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Discussion Overview

The discussion revolves around the concept of integration, specifically the interpretation of area under a curve and its significance in various contexts. Participants explore the meaning of area in mathematical terms, especially when units are not explicitly defined, and consider its applications in physical scenarios.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • BOAS questions the meaning of area under a curve without associated units, expressing confusion about how area can be discussed in purely mathematical terms.
  • One participant suggests that in the context of a force versus time graph, the area represents total momentum imparted to an object, which has defined units (Newton-seconds).
  • Another participant acknowledges the importance of units in physical examples but notes that in purely mathematical problems, areas may be discussed without physical significance.
  • There is a mention of hysteresis as an example of how area bounded by curves can have physical meaning, although the relevance of units is still questioned.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of units when discussing area. While some acknowledge that areas can have meaningful interpretations in physical contexts, others highlight that in purely mathematical scenarios, the absence of units can lead to confusion.

Contextual Notes

Some participants note that their mathematical problems do not represent physical quantities, leading to discussions about the implications of area without units. This raises questions about the foundational understanding of area in both mathematical and physical contexts.

BOAS
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hello,

i have a question about an explanation of integration as finding the area under a curve. I don't have any problems doing the integration but it's more a case of why?

Finding the area bound by two curves is easy enough but what does that area actually mean?

I know that it's a useful tool applied to physical examples where the area under the curve of something like radioactive decay has units, but I don't understand what it means to speak of area without giving it some units.

I hope that makes sense and that I have posted this in a relevant subsection. I don't know if it's a useful question to ask, but if there is an answer out there, I would be interested to know.

Thanks,

BOAS
 
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Hi BOAS! :smile:

Suppose you have a graph of force against time, for when a bat hits a ball.

The area under the curve is the total momentum imparted to the ball.

(and it equals ∫ F dt, which comes from F = d(momentum)/dt)
 
tiny-tim said:
Hi BOAS! :smile:

Suppose you have a graph of force against time, for when a bat hits a ball.

The area under the curve is the total momentum imparted to the ball.

(and it equals ∫ F dt, which comes from F = d(momentum)/dt)

Hello,

thanks for the response.

I do appreciate that integration as a method for finding the area beneath a curve is an extremely useful tool. What is bugging me a little, is the idea of finding the area bound by say, two curves. It doesn't seem meaningful to speak of area without units, although a quick sketch of the curve will tell you that there is indeed an area there.

It isn't really a big deal, but "area = 4" just feels odd and I was wondering if there was some explanation of why it can make sense without needing any units.
 
Hello BOAS! :smile:
BOAS said:
What is bugging me a little, is the idea of finding the area bound by say, two curves. It doesn't seem meaningful to speak of area without units, although a quick sketch of the curve will tell you that there is indeed an area there.

… why it can make sense without needing any units.

there are units …

eg in my example, the axes had units of Newtons and seconds, with an area of Newton-seconds

(and for an example of what an area-bounded-by-two-curves can represent, see http://en.wikipedia.org/wiki/Hysteresis :wink:)
 
tiny-tim said:
Hello BOAS! :smile:


there are units …

eg in my example, the axes had units of Newtons and seconds, with an area of Newton-seconds

(and for an example of what an area-bounded-by-two-curves can represent, see http://en.wikipedia.org/wiki/Hysteresis :wink:)

I do understand your example - I think I'm splitting hairs really.

In my maths class we've been doing problems that don't represent any physical thing, so neither axis had units.

I don't think it's especially important, I was just curious to see if there was a mathematical reason for being able to talk about are without units.
 

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