Clicker Question 1 What is the Simpson’s Rule estimate for the integral of f (x) = x2 + 1 on the interval [0, 4] using only two subdivisions? A. 12 2/3 B. 21 1/3 C. 25 D. 25 1/3 E. 27 2/3

Improper Integrals (9/30/13) 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. In both types, we’re trying to integrate over an unbounded region!

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.

Simplest 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

Clicker Question 2 What is ? A. Converges to 1 B. Converges to ½ C. Converges to 2 D. Converges to 4 E. Diverges

Clicker Question 3 What is ? A. Converges to 1 B. Converges to ½ C. Converges to 2 D. Converges to 4 E. Diverges

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

Global Behavior and Assignment With improper integrals of the first type, a rational function will behave like the ratio of its highest terms. Example: Does converge or diverge? Example: What about ? Assignment for Wednesday: Read Section 7.8 and do Exercises 1, 3, 5, 9, 11, 13, 17, 21, 31, 39, 49, and 51.