Infinite graph with finite area?

Click For Summary

Discussion Overview

The discussion explores whether a function can have an infinite graph while maintaining a finite area under the curve. Participants examine various functions and integrals, discussing convergence and properties of improper integrals.

Discussion Character

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

Main Points Raised

  • Some participants propose that a function must approach zero rapidly in both directions to have a finite area, citing examples like the error function and exponential decay functions.
  • Others discuss specific integrals, such as \(\int_{-\infty}^{\infty} e^{-x^2} dx\) and \(\int_{-\infty}^{\infty} \frac{1}{1+x^2} dx\), providing insights into their evaluation and convergence.
  • One participant introduces the concept of Gabriel's Horn, which has a finite volume but infinite surface area, as a related example.
  • There is a discussion about the Cauchy principal value of integrals, particularly for odd functions over symmetric intervals.
  • Some participants express confusion over the evaluation of integrals at infinity and seek clarification on specific techniques and theorems, such as Fubini's theorem.

Areas of Agreement / Disagreement

Participants express a range of views on the topic, with no clear consensus on the examples or methods discussed. Some agree on the properties of specific functions, while others challenge or refine these ideas, indicating ongoing debate.

Contextual Notes

Limitations include varying levels of mathematical knowledge among participants, with some unfamiliar with advanced concepts like Fubini's theorem or the definition of the error function. The discussion also reflects differing interpretations of convergence and the evaluation of improper integrals.

  • #31
Nope, it's not.
 
Physics news on Phys.org
  • #32
The reals are composed of numbers, infinity is not a number but rather a concept
 

Similar threads

Undergrad Infinite Series - Divergence of 1/n question
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Gabriel's Horn and Sum of Infinite Fractions: Contradiction or Connection?
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Convergence of an infinite series
  • · Replies 15 ·
Replies
15
Views
3K
High School Arithmetic mean of an infinite number of points?
  • · Replies 20 ·
Replies
20
Views
3K
Undergrad Infinite series and improper integrals
  • · Replies 3 ·
Replies
3
Views
2K
High School Area under a sine integral graph
  • · Replies 8 ·
Replies
8
Views
2K
High School Explain Integration to me, please
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad The Basic Area Problem (introduction to the topic of integrals)
  • · Replies 24 ·
Replies
24
Views
4K
Undergrad Area enclosed by a function involving 2 power towers
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad How to evaluate the enclosed area of this implicit curve?
  • · Replies 2 ·
Replies
2
Views
3K