1 Lecture 17: Improper integrals
1. Consider improper integrals at infinity.
2. Find areas as improper integrals.
3. The comparison theorem.
A question.
Can an region have infinite perimeter and finite area? If so, we can color it in, but
we cannot draw its outline.
Example. Find the area bounded by x= 1, y= 0 and y=e−x.
Solution. This is a new problem–until now, we have found the area of regions which
were of finite length in every direction.
We attempt to solve this problem, by first finding the area bounded by x= 1,
y= 0, y=e−xand x=N. This is
ZN
1
e−xdx =1
e−e−N.
If we take the limit as Ngoes to infinity, we expect to obtain a reasonable value for
the infinite area.
area = lim
N→∞
1
e−e−N=1
e.
Definition of improper integral.
Definition. If ais a real number and fis a function which is Riemann integrable on
each interval [a, N ] for N > a, then we define the improper integral as
Z∞
a
f(t)dt = lim
N→∞ ZN
a
f(t)dt.
If this limit exists and is finite, we say the improper integral is convergent. If the
limit is infinite or does not exist, we say the improper integral is divergent.
Exercise. Write out the corresponding definition for improper integrals at −∞.
Exercise. Re-read the definition of the Riemann integral Rb
af(t)dt and explain what
difficulties would arise if we let b=∞in this definition.
