Day 22 in MAT229

Today:

• Announcements:
  • Homework 7.7 is due tomorrow.
  • Your homework from 7.2 is returned:
    • 8 -- use the double angle formula
    • 24 -- factor a tan-squared
    • 44 -- Chris L.

• A few more examples of Improper Integrals (section 7.8)

  Improper integration forces us to revisit limits, which is probably a good thing. We'll start by taking a look at my summary of the section, for some of the definitions.

  • Motivation:
    • Improper Integration Handout
    • Type I: Infinite intervals
      Probability Density Functions (e.g. )
    • Type II: Discontinuous Integrands
      The class of Inverse power functions

  • Examples:
    • #1, p. 551
    • #5
    • #37
    • #53 (comparison)

• Section 8.1: Arc Length
  • What kinds of problems are we looking at in this section? Calculating the linear length of a curve (arc length).
    

  • Let's consider arc length:
    

    The usual trick is to discretize something continuous, and then pass to a limit:

    

    Now hopefully we can figure out how the formula will look:

    

    So what's the general formula?

    

  • Examples:
    • #2, p. 567
    • #3, p. 567
    • #37, p. 568