Exam 2 Wed Mar 1: 7pm-7:50pm in ??
Today: 7.8 Improper Integrals
Monday - president's day holiday (and almost my bday)
Next -- 11.1 sequences |
Example 5.7.1
Make sense of
![$ \int_{0}^{\infty} e^{-x} dx$](img855.png)
.
The integrals
make sense for each real number
![$ t$](img40.png)
. So consider
Geometrically the area under the whole curve is the limit
of the areas for finite values of
![$ t$](img40.png)
.
Figure 5.7.1:
Graph of
|
Example 5.7.2
Consider
![$ \int_0^1\frac{1}{\sqrt{1-x^2}} dx$](img859.png)
(see Figure
5.7.2).
Figure 5.7.2:
Graph of
|
Problem: The denominator
of the integrand tends to
0 as
![$ x$](img22.png)
approaches the upper
endpoint.
Define
Here
![$ t\to 1^-$](img864.png)
means the limit as
![$ t$](img40.png)
tends to
from
the left.
Example 5.7.3
There can be multiple points at which the integral is improper.
For example, consider
A crucial point is that we take the limit for the
left and right endpoints independently. We use
the point
0 (for convenience only!) to break
the integral in half.
The graph of
![$ \tan^{-1}(x)$](img872.png)
is in Figure
5.7.3.
Figure:
Graph of
|
Example 5.7.4
Brian Conrad's paper on impossibility theorems for elementary
integration begins: ``The Central Limit Theorem in
probability theory assigns a special significance to
the cumulative area function
under the Gaussian bell curve
It is known that
![$ \Phi(\infty) = 1$](img876.png)
.''
What does this last statement mean? It means that
Example 5.7.5
Consider
![$ \int_{-\infty}^{\infty} x dx$](img878.png)
.
Notice that
This diverges since each factor diverges independtly.
But notice that
This is
not what
![$ \int_{-\infty}^{\infty} x dx$](img878.png)
means (in this
course - in a later course it could be interpreted this way)!
This illustrates the importance of treating each bad point
separately (since Example
5.7.3) doesn't.
Example 5.7.6
Consider
![$ \int_{-1}^1 \frac{1}{\sqrt[3]{x}} dx$](img881.png)
.
We have
This illustrates how to be careful and break the function up into two
pieces when there is a discontinuity.
NOTES for 2006-02-22
Midterm 2: Wednesday, March 1, 2006, at 7pm in Pepper Canyon 109
Today: 7.8: Comparison of Improper integrals
11.1: Sequences
Next 11.2 Series
|
Example 5.7.7
Compute
![$ \int_{-1}^3 \frac{1}{x-2} dx$](img885.png)
.
A few weeks ago you might have done this:
This is not valid because the function we are
integrating has a pole at
![$ x=2$](img185.png)
(see Figure
5.7.4).
The integral is improper, and is only defined
if both the following limits exists:
![$\displaystyle \lim_{t\to 2^-} \int_{-1}^t \frac{1}{x-2} dx$](img887.png)
and
However, the limits diverge, e.g.,
Thus
![$ \int_{-1}^3 \frac{1}{x-2} dx$](img885.png)
is divergent.
Figure 5.7.4:
Graph of
|
Subsections
William Stein
2006-03-15