Day 23 in MAT229

Today:

• Announcements:
  • Your exam revisions are due tomorrow, at noon, in my office (443 MEP). You may certainly turn them in early!

  • Your quiz opportunity is over 8.1/8.2: arc length and surface area

• Improper Integration (section 7.8)
  • Motivation: Probability Density Functions (e.g. Example #9, p. 550)
  • Improper Integration Handout
  • Examples:
    • #37, p. 552 -- let's recap: so far, we
      • Replaced the integral with a limit;
      • Made a u-substitution
      • Did one integration by parts
      • Now are ready to use L'Hopital's rule to finish it off.

• 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 -- and then we'll use that in the next section, surface area calculations for areas of revolution).
    

  • Let's start with 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:
    • #37, p. 568
    • #3, p. 567
    • #25
    • #21

• Section 8.2: surface area (for areas of revolution)
  

  We discretize bits of the surface, and then make a series of little bands; take the limit, and get the general formula:

  

• Examples:
  • #19, p. 574
  • #25
  • #28
  • #33