- Homework for 8.1 due (those who handed it in last time get a bonus "homework drop" -- that's your reward for being so brave).
- What are "homework drops"?
- Homework for 8.6 is due next time.
- Read Section 9.4 for next time, responding to the worksheet for section 9.4;
- Prepare for a quiz next time on the content of section 9.4.
- You can't say "the limit" and expect me to understand!
- We need to be more precise (less mysterious).
- In this section we deal with integrals of two sorts: those whose limits of integration are infinite, and those with vertical asymptotes.
- Definition concerning the first type:
Here's an important example:
- The following figure illustrates the first type, and also the
second:
The integral exists over
But what can we say about the integral over
:
- There's a general rule for powers, as we saw last time (it's easy to take the limits
and check):
Last time we checked in this case:
As noted last time, the case of p=1 is interesting, and we can use symmetry to see that the area is unbounded in both cases.
Last time we checked that these work using a handout. Let's do a few more, to get warmed up for today's new material: convergence.
Examples:
- #13, p. 464
- #12, p. 464
- #43, p. 465
- There's a third component of this section which is interesting:
the idea of using bounding to deduce whether an integral
diverges or converges. For example, we can use comparison to
show that, in fact, as the caption says, the area under the following
curve is finite:
This one has infinite limits on both ends -- how do we handle that?- We could use symmetry, of course; but more generally,
How would we use comparison? We find a function which is everywhere greater than the function but which has finite area. Can you think of a good candidate?
Let's use my choice and this handout. What are the issues?
- We could use symmetry, of course; but more generally,
- Examples:
- #60, p. 465
- #85 and 86, p. 466
