Day 16 in Mat122

Today:

• Hand in 6.1 with the exam -- I'll promise to pull one off the homework, essentially.

• First exam, tomorrow!

• Section 6.2: Setting up Integrals
  • The hardest job is often visualizing what's going on. Then the problem becomes figuring out the little chunks of stuff that one wants to add up to get the whole quantity of stuff. Here are some classic examples:

  • Volumes:
  • Population density:
  • Flow rate:
  • Average Value:
  • Mass density:

  • Examples:
    • #2, p. 317
    • #4abc, p. 317
    • #6, p. 317
    • #26, p. 318
    • #30, p. 319
    • #33, p. 319
    • #53, p. 319

• Section 6.3: Volumes of revolution
  • The definition:

    

    More generally...

  • We may want to consider two functions simultaneously, to give a "hollow" volume of revolution:

    

  • We can rotate about an axis other than the x-axis (e.g. the y-axis!), or even
    • the line x=-2, or
    • the line y=12 or y=15

  • Examples:
    • #2, p. 325
    • #15, p. 325
    • #32, p. 326
    • #38, p. 326
    • #40, p. 326