Lecture 39
Improper Integral
In this lecture we will study
•Integrals over Infinite Interval
•Integrals whose Integrands become Infinite
Integrals over Infinite Interval
• As we saw be fore, for a given conti nuous functi on f,
the definite integral is
It is assumed that the interval [a,b] is finite.
• W hat if we look at [a, +¥) and the corresponding
integral
ò
b
a
dxxf )(
ò
+¥
a
dxxf )(
• In this case we define what is called an improper
integral over an infinite interval.
What does it mean to integrate all the
way to + ¥?The answer will be clear if we define
this integral as a limit in the following way.
What this does is to first turn the integral into
more familiar form of over a finite interval, and then
we let the upper limits of the interval approach 0
and see what happen to the answer we had got
earlier.
òò +¥®
+¥
=
l
a
l
a
dxxfdxxf )(lim)(
• If this limi ts exists, t hen we say that t he Improper
Integral Converges, and the value of the limit is
assigned to the integral.
• If the limit does not exist, then we say that t he
Improper Integral Diverges, and no finite value is
assigned.
• Lets get some geometric ideas t o understand
things.
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