Integral of Bell Curve: Area of Slices

  • Context: Undergrad 
  • Thread starter Thread starter icystrike
  • Start date Start date
  • Tags Tags
    Area
Click For Summary

Discussion Overview

The discussion revolves around the integral of the function \( e^{-r^{2}} \) and its interpretation in relation to the area of slices of a bell curve, particularly in the context of rotation about the z-axis. Participants explore the implications of this integral in terms of area and its applications.

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions the area represented by the integral \( \int_{-\infty }^{\infty }e^{-r^{2}}dr \), suggesting it relates to slices of the bell curve.
  • Another participant clarifies that the integral represents a vertical slice of the bell curve through its center.
  • A different participant notes that calculating an integral does not necessarily imply finding an area, highlighting the broader applications of integrals.
  • One participant mentions their teacher's description of the slice as parallel to the y-axis, while they believe it should pass through the origin, suggesting a potential misunderstanding.
  • Another participant proposes that the teacher's description may still yield the same shape, albeit with a different factor.
  • A later reply expresses gratitude for clarification on the concept, indicating a desire to ensure understanding.

Areas of Agreement / Disagreement

Participants express differing interpretations of the integral's representation and the nature of the slices, indicating that multiple competing views remain without a clear consensus.

Contextual Notes

There are unresolved assumptions regarding the definitions of the slices and the implications of the integral in various contexts, including potential misunderstandings in the teacher's explanation.

icystrike
Messages
444
Reaction score
1
Can i ask what is the area we are referring to when we take such integral (r is the radius):

[tex]\int_{-\infty }^{\infty }e^{-r^{2}}dr[/tex]

I'm suspecting that its is the area of slices of bell curve that rotates about the z-axis.

attachment.php?attachmentid=33094&stc=1&d=1300169059.png
 

Attachments

  • graph.png
    graph.png
    6.6 KB · Views: 443
Last edited:
Physics news on Phys.org
hi icystrike! :smile:

it's the integral of a vertical slice of the bell curve through its centre :wink:
 
You understand, I hope, that finding area is one possible application of the integral. When we calculate an integral we are not necessarily finding any area at all!
 
Thanks tiny-tim and HallsofIvy!

Yes! I know that! We can use Integral to compute things like work, flux, centroids .. =D

Its just that my teacher actually relate the slice as "some slice that is parallel to the y-axis" while i think that it should be the slice that is passing through origin(He've probably made some mistake)... (My teacher was actually comparing the volume of a rotated bell curve about z axis by slice and shells to evaluate the area under bell curve - [tex]A^{2}=\pi[/tex] )
 
hi icystrike! :smile:
icystrike said:
Its just that my teacher actually relate the slice as "some slice that is parallel to the y-axis" while i think that it should be the slice that is passing through origin

i think he means that it'll be the same (it's the same shape), apart from a factor e-x2 :wink:
 
Oh! Thats what he meant! Truly enlighten! Thanks Tim! :)
(Came to ensure that i get the concept right)
 

Similar threads

Undergrad How to evaluate the enclosed area of this implicit curve?
  • · Replies 2 ·
Replies
2
Views
3K
High School Calculating shaded area: I'm getting discrepancies between methods
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad The Basic Area Problem (introduction to the topic of integrals)
  • · Replies 24 ·
Replies
24
Views
4K
Undergrad Question about Integrals to Determine Volume vs Surface Area
  • · Replies 5 ·
Replies
5
Views
4K
High School Integration from "Area Under Curve" Perspective: Explained
  • · Replies 11 ·
Replies
11
Views
2K
High School Question about the limits of integration in a change of variables
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Why does the integral of sine of x^2 from - infinity to + infinity diverge?
  • · Replies 4 ·
Replies
4
Views
3K
High School Why is this definite integral a single number?
  • · Replies 20 ·
Replies
20
Views
4K
Undergrad Negative area above x-axis from integrating x^2?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Having trouble understanding a seemingly simple u-substitution
  • · Replies 5 ·
Replies
5
Views
2K