Section 7.8 Improper Integrals
We've been integrating and have been using the Fundamental Theorem of Calculus that says we can find an antiderivative and drop in the bounds...but it also requires us to be dealing with a continuous function.
What if one of the bounds is a vertical asymptote for the function?
What does this integral mean now?
What if the bounds are not numbers? What if the function has a horizontal asymptote and we allow x to roll off to infinity in both directions, or in one direction?
What does this integral mean now?
We've actually dealt with something like this earlier this term when we talked about the Cumulative Area Function. We integrated a function but we let the upper bound move so that we created an area function and we could pick any upper bound we want.
By thinking through the limit we see that there is a cap to the area. No matter how far out we let x go the area accumulated will top out at 0.5. The area is finite.
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When there is a finite area we say the integral "converges." If the area just grows without bound then the integral "diverges."
Type 1 Improper Integrals: These are "easy" to spot. You will see infinity as one of the bounds. Set these up as limits and process the integral. Evaluate the limit last.
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The Type 2 Improper Integrals are not so easy to spot.
From now on, take a look at the function and note the domain. If the bounds of the integral plow right over an asymptote then you need to set the integral up as a limit.
If one of the bounds is a vertical asymptote for the function, then you need to set up the integral up as a limit.
Example:
If we ignore the continuity requirement that comes with the Fundamental Theorem of Calculus, we can integrate and we will get a number...but if you look at a graph you will see that our number doesn't make any sense.
Now look at the graph. Note the entire function is above the x­axis so any area would be positive. Then notice that we integrated right over a point of discontinuity...that's bad.
So we will break this function at the vertical asymptote and write both integrals as limits. Notice one limit approaches 1 from the left and the other approaches 1 from the right.
We integrated only one of the integrals and see it diverges. We don't need to worry about the other side. So...with vertical asymptotes, write the integral as a limit.
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