There are two ways to extend the Fundamental Theorem of Calculus. One is to use an infinite interval , i. One of the most important applications of this concept is probability distributions because determining quantities like the cumulative distribution or expected value typically require integrals on infinite intervals. To compute improper integrals, we use the concept of limits along with the Fundamental Theorem of Calculus.
Since we are dealing with limits, we are interested in convergence and divergence of the improper integral. If the limit exists and is a finite number, we say the improper integral converges. Otherwise, we say the improper integral diverges , which we capture in the following definition. First we compute the indefinite integral. This is an integral over an infinite interval that also contains a discontinuous integrand.
It is important to remember that all of the processes we are working with in this section so that each integral only contains one problem point. In order for the integral in the example to be convergent we will need BOTH of these to be convergent.
If one or both are divergent then the whole integral will also be divergent. We know that the second integral is convergent by the fact given in the infinite interval portion above. So, all we need to do is check the first integral. Notes Quick Nav Download. You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width.
Example 1 Evaluate the following integral. Example 3 Determine if the following integral is convergent or divergent. If it is convergent find its value. Example 4 Determine if the following integral is convergent or divergent. Example 5 Determine if the following integral is convergent or divergent. Example 6 Determine if the following integral is convergent or divergent. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more.
How to determine whether an integral is convergent Ask Question. Asked 8 years, 7 months ago. Active 8 years, 5 months ago. Viewed 8k times. Add a comment.
Active Oldest Votes. Ron Gordon Ron Gordon k 16 16 gold badges silver badges bronze badges.
0コメント