Improper Integrals (9/24/08) There are two types of “improper integrals”: First Type: Definite integral taken over a ray or the whole real line, rather than over an interval of finite length. Second Type: Definite integral of a function which becomes unbounded (i.e., “blows up”) on the interval of integration.

Example of First Type What is By this we mean, what is If this limit is a finite number, then we say the integral converges. If it is not a finite number, we say the integral diverges.

Example of the second type What is As before, what this means is We make the same definition of converges and diverges

Solutions of the examples

Simpler approach In practice, we usually dispense with the “limit as b goes to” and simply use  or 0. We understand that  in the numerator will cause divergence, as will 1/0, whereas  in the denominator gives a value of 0 to that term. For example, then

Global Behavior With improper integrals of the first type, a function will behave like the ratio of its highest terms. Example: Does converge or diverge? Example: What about ?

Basic Facts on Power and Exponential Functions converges for p > 1 and diverges for p  1. converges for p < 1 and diverges for p  1. converges for all a > 0.

Assignment for Friday Hand-in #1 is due tomorrow (Thurs) at 4:45. Read Section 7.8 and do Exercises 1, 3, 5, 9-17 odd, 21, 27, 31, and 39.