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We'd like to introduce a couple of new words to help us talk about limits. If you're rusty on how limits work, we recommend reviewing them.
When a limit exists and equals L, we say that limit converges to L. The phrase "converges to" means the same thing as the word "approaches."
converges to 0.
Sometimes we say a limit converges without bothering to say its value.
When a limit doesn't exist, we say that limit diverges. The limit
These integrals are accounting for the area between the graph of and the x-axis on intervals whose right endpoint is 1 and whose left endpoints are moving closer and closer to 0:
As b approaches 0, the area
approaches the total area between the graph of and the x-axis on (0, 1].
In symbols, the total area between and the x-axis on (0,1] is
We abbreviate this limit by
looks like a normal definite integral, it isn't, because the integrand has an asymptote at one of the endpoints, x = 0.
Since the function is undefined at x = 0, there's something weird about
We aren't looking at a normal definite integral here. There's something improper about it, which leads to our next important definition.
Improper integrals are limits of definite integrals. The integrals
are examples of improper integrals.
There are two types of improper integrals. In the first type, the limits are badly behaved (that is, ∞ or -∞). Such integrals would look like one of these (c is a constant):
In the second type, the functions are badly behaved. These integrals will look like normal definite integrals
but somewhere in the interval from a to b a vertical asymptote will be lurking, as the function zooms off to infinity!
Be Careful: Improper integrals are limits. As with all limits, improper integrals may converge or diverge—that is, they may or may not exist.
From now until the end of calculus, whenever you're asked to evaluate an integral, first ask yourself if that integral is improper. Just because the expression
is written down, it doesn't mean that expression has a numerical value!
Even though improper integrals are limits, we still think of them as areas. The improper integral
is the area between and the x-axis on (0,1].
The improper integral
is the area between and the x-axis on [1,∞).
When an improper integral of a non-negative or non-positive function diverges, it means the area described is infinite.
Now let's look at the two types of improper integrals in a little more depth.
Improper integrals with badly behaved limits are integrals where one or both of the limits is infinite.
These integrals look like
If only one limit of integration is infinite then the other limit of integration, c, would be a finite constant.
Remember that writing ± ∞ for the limits of integration is a shortcut. These improper integral creatures are really limits.
To evaluate this type of improper integral, we have to work out the integral inside the limit and then evaluate the limit.
When only one of the limits of integration is infinite we can work things out directly. When both limits of integration are infinite we have to split the integral into two pieces first.
There's some weird stuff going on here. If we take a value of p > 1, the integral
converges. But if we take a value of p < 1, that same integral diverges. And the graphs don't even look that different!
Take, for example, p = 2 and :
One way to explain this is that the graph of is able to bound a finite area above the x-axis on [1,∞) because it gets close to zero more quickly than the graph of .
There's another thing going on here we need to point out. Let
Then the function f (x) converges to 0 as x approaches ∞. In symbols,
However, the integral
diverges. It's possible for the function to converge to 0 at ∞, but for an improper integral of the function, with an upper limit of ∞, to diverge.
However, if the function f (x) diverges at ∞, then there's no hope for the integral
to converge. No matter what c is, if f zooms off to ± ∞ or can't make up its mind what to equal as x approaches ∞, then
is going to diverge.
In fact, if
equals anything other than 0, even if the limit converges, then
will diverge. If then the function and the x-axis will enclose an infinite area on the interval [c, ∞).
With the other type of improper integral, it's possible for a function to diverge while the improper integral converges.
Be Careful: When you talk about the convergence or divergence of something, make sure you say what is doing the converging or diverging. Saying that a function converges and saying that an integral converges are not the same thing.
These ideas of convergence and divergence will come up again when we're talking about sequences and series.
So far we've looked at integrals with only one infinite limit. If both limits are infinite, we need to break up the integral somewhere in the middle.
where c is any real number you like (we usually pick something easy like 0 or 1, depending on the function). Since each of the integrals on the right-hand side is really a limit in disguise, this equation says
A sum of limits only exists if both limits involved exist. This is both good news and bad news.
The good news is that if you go to work out
and you find that either
diverges, then you're done. The integral
The bad news is that if
does exist, you have to work out both of the limits
and then add them up. This comes with its own good news, though. You'll already have found the antiderivative for f (x) after the first limit, so the second limit won't be as much work.
Improper integrals with badly-behaved functions are deceptive.
They look like normal definite integrals
but somewhere in the interval from a to b, possibly at one of the endpoints, there will be a vertical asymptote where the function f tends towards ∞ or -∞.
In fact, the function may have more than one vertical asymptote in the interval of integration.
Because of all these possibilities for bad behavior, we recommend graphing functions and looking at their formulas before trying to integrate. Be on the lookout for bad behavior! We want to catch all the places where the function is badly-behaved.
To evaluate this type of improper integral, the first thing we have to do is figure out where the function is badly behaved. Then we can write the improper integral as the limit it really is, work out the integral inside the limit, and finally evaluate the limit.
We'll start with improper integrals where the function is badly behaved at one limit of integration, but nowhere else.
Every other improper integral with a badly-behaved function can be broken down into integrals like this.
There's weird stuff going on again. If we take p > 1, the integral
diverges. If we take p < 1, the integral converges. This is the opposite of what happened with the other type of improper integral.
To take an example, on the interval [0,1] the area between and the x-axis is infinite but the area between and the x-axis is finite. The graphs don't even look that different. It's perfectly normal if this makes your brain bend a little.
When talking about things "converging" or "diverging," make sure you say whether you're talking about a function or an integral. The function
diverges as x approaches 0. However, the integral
converges. This means we can have a function diverge but a closely associated improper integral diverge. It's also possible for both the function and an improper integral to diverge. The function diverges as x approaches 0, and so does its improper integral
If a function is badly-behaved in between its limits of integration, we split the improper integral into two pieces. We do this by making a new endpoint at the value of x where the function is badly-behaved. Suppose that
is an improper integral because f has a vertical asymptote at x = b:
Then we split the integral into two new improper integrals:
The function f is badly behaved at just one limit of integration for each of these improper integrals.
When we do this, it's important to remember that
are both improper integrals, and thus are both limits. Both of these limits must exist in order for
to exist. Said another way, if either
There's both good news and bad news here. The good news is that if we find one of the new improper integrals diverges, we're done with the problem. The bad news is that if both of the new improper integrals converge, we have to work out both of them. This is the same good news / bad news we got for the other type of integral.
If you're given some random integral to integrate, you probably won't be told whether it's improper or not. It might be improper because of badly behaved limits, a badly behaved function, or both. Either way, you can break it into smaller improper integrals, each of which is improper for only one reason.