Intuition of improper integrals of type II

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Discussion Overview

The discussion revolves around the intuition behind improper integrals of type II, particularly focusing on the integration of functions that are continuous on an interval but have discontinuities at one endpoint. The example provided is the integral of \( \frac{1}{\sqrt{x}} \) from 0 to 1, raising questions about the nature of the area represented by the integral and how to reconcile the concept of infinite heights of rectangles with finite areas.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about how to intuitively understand the area under a curve when some rectangles approaching the lower limit seem to have infinite heights, questioning how these can contribute to a finite area.
  • Another participant clarifies that the rectangles have heights approaching infinity rather than being of infinite length, prompting further exploration of limits and products of functions as they approach infinity.
  • A side note is introduced regarding the definition of continuity, suggesting that since the function is not defined at \( x=0 \), it may not be appropriate to classify it as discontinuous at that point.
  • Further discussion questions whether a function \( g(x) \) can decrease sufficiently fast to keep the product \( f(x)g(x) \) finite, paralleling the idea of how bases of rectangles can decrease while heights approach infinity.

Areas of Agreement / Disagreement

Participants express differing views on the nature of discontinuity at the lower limit of integration and the implications for understanding improper integrals. There is no consensus on the intuition behind the area calculation or the definitions being discussed.

Contextual Notes

The discussion highlights potential limitations in understanding continuity in relation to the domain of functions and the behavior of limits, particularly in the context of improper integrals.

Bipolarity
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Let's say you have a function that is continuous in (a,b] but discontinuous at x=a and you integrate it from a to b.

For example, \int^{1}_{0} \frac{1}{\sqrt{x}}dx

I understand that the integral exists, and it can be easily computed by using the limit as x approaches 0 from the positive side.

What bothers me is the intuition of the problem. If we divide the area up into infinite rectangles, it seems that some of the rectangles approaching the lower limit are of infinite length, so that their area must also be infinite? I can't fathom how you can add up these rectangles that approach lengths of infinity.

BiP
 
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Bipolarity said:
it seems that some of the rectangles approaching the lower limit are of infinite length

A clearer view is that some of the rectangles approaching the lower limit "have heights that are approaching infinite length", not that they "are" of infinite length.

What's your intuition about \lim_{x \rightarrow a} f(x) g(x) when \lim_{x \rightarrow a} f(x) = \infty ? Do you think a g(x) can get small "fast enough" to make the product f(x)g(x) stay finite?
 
Stephen Tashi said:
A clearer view is that some of the rectangles approaching the lower limit "have heights that are approaching infinite length", not that they "are" of infinite length.

What's your intuition about \lim_{x \rightarrow a} f(x) g(x) when \lim_{x \rightarrow a} f(x) = \infty ? Do you think a g(x) can get small "fast enough" to make the product f(x)g(x) stay finite?

Why not?

BiP
 
Just a side note, but in the example you gave above, can we really say the function is discontinuous at x=0? In this case x=0 is simply not part of the domain. It's my understanding that we can only really think of continuity of a function in terms of regions it's defined upon. Perhaps this has something to do with the answer?
 
Bipolarity said:
Why not?

BiP

Why not what? If you think g(x) can decrease quickly enough so that \lim_{x \rightarrow a} f(x) g(x) is finite, then what problem do you have beleiveing that the bases of rectangles can decrease "fast enough" so that the areas of the rectangles remains finite even though the height approaches infinity?

As to the definition of continuity, the usual definition is "A real valued function f(x) whose domain is a subset of the real numbers is continuous at x = a if and only if \lim_{x \rightarrow a} f(x) = f(a)". Hence the definition requires that f(a) exists. Hence if f(x) does not exist at x = a then f(x) is not continuous at x = a.
 

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