

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.
Replies: 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
hawley
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.
asmith
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).
tee
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Update: June 2012

