Department of Energy Argonne National Laboratory Office of Science NEWTON's Homepage NEWTON's Homepage
NEWTON, Ask A Scientist!
NEWTON Home Page NEWTON Teachers Visit Our Archives Ask A Question How To Ask A Question Question of the Week Our Expert Scientists Volunteer at NEWTON! Frequently Asked Questions Referencing NEWTON About NEWTON About Ask A Scientist Education At Argonne Intuitive reason for an integral that does not exist
Name: yendor
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995

"Find an example of a function f(x) defined on [0,1] such that the definite integral from 0 to 1 of f(x)dx does not exist. Give an intuitive reason why the integral does not exist."

My math professor asked me this question, or one similar to it, and it got me too wondering. Please help me out.

Assuming the integral does not simply diverge, then you are probably talking about pathological functions. Two simple examples are:

f(x) = sin(1/x)
f(x) = { 0 if x is rational, 1 if x is irrational


Generally, for an integral not to exist it must be that somewhere the function being integrated produces arbitrarily large negative and positive pieces, and it is not clear how they should compensate one another (infinity minus infinity is not defined). A simple example defined on the interval [-1,1] is the function f(x) = 1/x. Maybe the integral should be zero, but it really is not well defined.


Neither of the previous answers is very satisfactory. sin(1/x) and 1/x are not defined on the intervals stated and the rational irrational is not in the intuitive category. Perhaps a better example is

f(x)=1/x^2 for 0infinity as

the partition is refined. So, the limit defining the integral does not exist (area under the curve is unbounded).


Click here to return to the Mathematics Archives

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (, or at Argonne's Educational Programs

Educational Programs
Building 360
9700 S. Cass Ave.
Argonne, Illinois
60439-4845, USA
Update: June 2012
Weclome To Newton

Argonne National Laboratory