1. 10.3 Integrals with Infinite Limits of Integration
Rita Korsunsky

2. Integrals with Infinite Limits of Integration are Improper Integrals
Definition:
i) If f is continuous on [a, ), then
ii) If f is continuous on (-, a], then
provided the limit exists.
An improper integral is said to converge if the limit exists, and if it does not exist, the integral is said to diverge.

3. Example 1
Determine whether the integral converges or diverges, and if it converges, find its value.
(converges)
(diverges)

4. If area under a curve is infinite (integral diverges), then volume of solid of revolution when you rotate this region around x or y-axis does not have to be infinite. (Integral could converge)
Diverges from example 1 part b
Converges from example 1 part a

5. Diverges from example 1 part b
If area under a curve is a constant (integral converges), then volume of solid of revolution when you rotate this finite region around x or y-axis does not have to be constant. (Integral could diverge)
Converges from example 1 part a
Diverges
Diverges from example 1 part b
Converges

6. Example 2
Assign an area to the region that lies under the graph of y=ex, over the x-axis, and to the left of x=1.

7. Another Form of Improper Integral
Let f be continuous for every x. If a is any real number, then
provided both of the improper integrals on the right converge.

8. Example 3 Evaluate the following: Similarly: Add both values:
First evaluate this: