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.
For our usual definite integrals
the function f ( x ) must be continuous on the interval from a to b. This includes continuity at the endpoints a and b.
Since the function is undefined and therefore discontinuous 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.
For the function f graphed below,
is the area between f and the x-axis on [1,∞). This area is certainly infinite!
Now let's look at the two types of improper integrals in a little more depth.