Improper Integrals

Improper Integrals - Improper Integrals In this section we...

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Improper Integrals In this section we need to take a look at a couple of different kinds of integrals. Both of these are examples of integrals that are called Improper Integrals. Let’s start with the first kind of improper integrals that we’re going to take a look at. Infinite Interval In this kind of integrals we are going to take a look at integrals that in which one or both of the limits of integration are infinity. In these cases the interval of integration is said to be over an infinite interval. Let’s take a look at an example that will also show us how we are going to deal with these integrals. Example 1 Evaluate the following integral. Solution This is an innocent enough looking integral. However, because infinity is not a real number we can’t just integrate as normal and then “plug in” the infinity to get an answer. To see how we’re going to do this integral let’s think of this as an area problem. So instead of asking what the integral is, let’s instead ask what the area under on the interval is. We still aren’t able to do this, however, let’s step back a little and instead ask what the area under is on the interval were and t is finite. This is a problem that we can do. Now, we can get the area under on simply by taking the limit of A t as t goes to infinity.
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This is then how we will do the integral itself. So, this is how we will deal with these kinds of integrals in general. We will replace the infinity with a variable (usually t ), do the integral and then take the limit of the result as t goes to infinity. On a side note, notice that the area under a curve on an infinite interval was not infinity as we might have suspected it to be. In fact, it was a surprisingly small number. Of course this won’t always be the case, but it is important enough to point out that not all areas on an infinite interval will yield infinite areas. Let’s now get some definitions out of the way. We will call these integrals convergent if the associated limit exists and is a finite number ( i.e. it’s not plus or minus infinity) and divergent if the associated limits either doesn’t exist or is
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Improper Integrals - Improper Integrals In this section we...

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