Is the Limit Definition of a Definite Integral Correct?

  • Thread starter Thread starter GreenPrint
  • Start date Start date
  • Tags Tags
    Integration
Click For Summary

Homework Help Overview

The discussion revolves around the limit definition of a definite integral, specifically questioning the validity of the expression involving limits as the bounds approach the endpoints of the interval. The subject area is calculus, focusing on integrals and their definitions.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the formulation of the limit definition and question the arguments of the limits involved. There is an examination of the conditions under which the definition holds true, particularly regarding the continuity of the function involved.

Discussion Status

The discussion is ongoing, with some participants affirming the validity of the limit definition under certain conditions. There are differing perspectives on the implications of continuity and the nature of the integral being discussed.

Contextual Notes

Participants are considering both proper and improper integrals, noting the necessity of limits in the latter case. There is a focus on the implications of the function's continuity on the validity of the limit definition.

GreenPrint
Messages
1,186
Reaction score
0
\int^{b}_{a} f(x) dx = lim_{c → a^{+}} lim_{d → b^{-}} \int^{d}_{c} f(x) dx

Is this true?
 
Physics news on Phys.org
GreenPrint said:
\int^{b}_{a} f(x) dx = lim_{c → a^{+}} lim_{d → b^{-}} \int^{d}_{c} f(x) dx

Is this true?

What are the arguments of the limit in this case?
 
HACR said:
What are the arguments of the limit in this case?

a and b on the left hand side and I replaced these with c and d evaluated with limits with a and b, a from the right and b from the left.
 
Yes it always true, either if f(x) is continuos on [a,b] or on (a,b). In the first case you have a proper integral, in fact the primitive F(x) is also continuos and the limit is the value of F at a (or b). In the second case you have an improper integral and in that case is necessary to use the limit to check whether or not the result is a finite value.
 
hmmm...
ya i guess so thanks
 

Similar threads

Prove that the integral is equal to ##\pi^2/8##
  • · Replies 105 ·
4
Replies
105
Views
10K
Limit and Integration of ##f_n (x)##
  • · Replies 8 ·
Replies
8
Views
2K
Integrating an inequality for two functions of the same variable
  • · Replies 9 ·
Replies
9
Views
2K
Determine limits of integration in double integral change of variables
  • · Replies 2 ·
Replies
2
Views
2K
Find the value of the definite integral
  • · Replies 8 ·
Replies
8
Views
2K
Dirac delta function approximation
  • · Replies 13 ·
Replies
13
Views
2K
Solve the given problem involving integration
  • · Replies 6 ·
Replies
6
Views
2K
Integrate [cosec(30°+x)-cosec(60°+x)] dx in terms of tan x
  • · Replies 3 ·
Replies
3
Views
2K
Limit Definition of Derivative as n Approaches Infinity
  • · Replies 20 ·
Replies
20
Views
3K
Is this a valid representation of a derivative?
  • · Replies 12 ·
Replies
12
Views
1K