# Improper Integrals

 Exam 2 Wed Mar 1: 7pm-7:50pm in ?? Today: 7.8 Improper Integrals Monday - president's day holiday (and almost my bday) Next -- 11.1 sequences

Example 5.7.1   Make sense of . The integrals

make sense for each real number . So consider

Geometrically the area under the whole curve is the limit of the areas for finite values of .

Example 5.7.2   Consider (see Figure 5.7.2).
Problem: The denominator of the integrand tends to 0 as approaches the upper endpoint. Define

Here means the limit as tends to from the left.

Example 5.7.3   There can be multiple points at which the integral is improper. For example, consider

A crucial point is that we take the limit for the left and right endpoints independently. We use the point 0 (for convenience only!) to break the integral in half.

The graph of is in Figure 5.7.3.

Example 5.7.4   Brian Conrad's paper on impossibility theorems for elementary integration begins: The Central Limit Theorem in probability theory assigns a special significance to the cumulative area function

under the Gaussian bell curve

It is known that .''

What does this last statement mean? It means that

Example 5.7.5   Consider . Notice that

This diverges since each factor diverges independtly. But notice that

This is not what means (in this course - in a later course it could be interpreted this way)! This illustrates the importance of treating each bad point separately (since Example 5.7.3) doesn't.

Example 5.7.6   Consider . We have

This illustrates how to be careful and break the function up into two pieces when there is a discontinuity.

 NOTES for 2006-02-22 Midterm 2: Wednesday, March 1, 2006, at 7pm in Pepper Canyon 109 Today: 7.8: Comparison of Improper integrals 11.1: Sequences Next 11.2 Series

Example 5.7.7   Compute . A few weeks ago you might have done this:

This is not valid because the function we are integrating has a pole at (see Figure 5.7.4). The integral is improper, and is only defined if both the following limits exists:

and

However, the limits diverge, e.g.,

Thus is divergent.

Subsections
William Stein 2006-03-15