## Lecture Notes: Monday, November 30, 1998

Twelfth Homework
Assignment - for discussion at recitations on Tuesday December 8 and
Thursday December 10.

### Chapter 7 **Section 4 (Improper Integrals)**

Integrals so far: Only have treated bounded functions on finite
intervals.

"**Improper**" Cases

- unbounded interval.
- Ex: integral of 1/x^2 from 1 to plus infinity
(convergent)
- Ex: integral of 1/x from 1 to plus infinity
(divergent)
- general case

- Ex: integral of x*exp(-x^2) from - to + infinity (plot in
MAPLE)
- general case

- unbounded functions (not in text)
- Ex: integral from 0 to 1 of 1/sqrt(x)
- general case

**Caution: **

- Ex: integral of 1/x^2 from -1 to 1. The answer is NOT 2. This
integral is divergent at x = zero.

**Interesting Example:**

- Area under 1/x from 1 to plus infinity is infinite
- But the volume of the solid of revolution generated by
revolving 1/x about the x-axis from 1 to plus infinity is finite.
- It turns out that the surface area of this figure is given
by the integral from 1 to plus infinity of (2*Pi/x)*sqrt(1 +
1/x^4). This is larger than the integral of 2*Pi/x. Since the
latter is infinite, so is the former. Thus here is a figure
with infinite surface area that "can be painted with a finite
amount of paint" (since the volume of the inside is
finite).

Examples: briefly mentioned #44 (did not have time for #8, 12,
36)

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