## 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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