28 - Improper Integrals
So you obviously know how to perform definite integration by now.
Remember how we said that integration is essentially taking the area under the curve?
But what happens when we want to take the integral of an infinitely large graph?
Consider the graph of
Take a look as to what happens as we take larger and larger x.
The area under the curve will get larger and larger. Infinitely larger, in fact!
-(1).gif)
So, what would happen in this case?
Well, let's suppose that we had to integrate with an infinite upper bound.
Let's assume that this was a finite bound first. Let's suppose that this
Our integral would be:
AHA! We can do this as a limit!
Suppose that instead of this a you now want an infinity. An infinity isn't a number, but you can perform this calculation as a limit to infinity!
And that's precisely what the improper integral is.
When we have a boundary to infinity, or to a point where a function has a vertical asymptote, the improper integral is defined: