Day 19 in MAT229

Last time: Exam review

Next time: Section 8.6/Section 9.1

Today:

Announcements
- Exam curve: I'm going to add 15 points to your scores
- If you'll come and see me I'll give you back 20% of your lost points.
  And if you have special concerns, you should come see me, and soon.
I'm returning 7.8 problems (graded 83, which we discussed in class):

${\theta(x) =\tan^{-1}(\frac{b+h}{x})-\tan^{-1}(\frac{h}{x})}$

${\theta'(x) =\frac{h}{x^2+h^2}-\frac{b+h}{x^2+(b+h)^2}}$
And setting ${\theta'(x) =0}$ yields

${x=\sqrt{bh+h^2}}$
Section 8.2 is due today.

Let's wrap up Section 8.6: Improper integrals

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 ${[2,\infty)}$ But what can we say about the integral over (0,2]?
There's a general rule for powers (it's easy to take the limits and check):

Let's check in this case:

The case of p=1 is interesting, and we can use symmetry to see that it is so.
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,

This one has infinite limits on both ends -- how do we handle that?
- We could use symmetry, of course; but more generally,
  
  ${\int_{-\infty}^{\infty}f(x)dx=\int_{-\infty}^{0}f(x)dx+\int_{0}^{\infty}f(x)dx}$
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?
Examples:
- #15, p. 464
- #12, p. 464
- #45, p. 465
- #98, p. 467 (see note preceeding exercise #96)

Website maintained by Andy Long. Comments appreciated.