Dave's Web Corner

MTH-261 Week 8 (October 13-19)

recommended exercises
- 3, 5, 9, 11, 13, 15
learning objectives
- recognize when a function of two variables is integrable over a rectangular region
- recognize and use the properties of double integrals
additional notes
- the purpose of integration
  - the length/area and area/volume dualities
  - the contribution of a function to a length, area, or volume
- the three components of an integral ∫_I f dx, ∫∫_R f dA, ∫∫∫_G f dV
  - the limits: I, R, G
  - the integrand: f
  - the order of integration: dx, dA, dV
- evaluating integrals geometrically
- riemann sums and the double integral
- average value and "net" value

recommended exercises
- 3, 7, 11, 17, 21, 23, 29, 31, 33, 35, 37
learning objectives
- evaluate a double integral over a rectangular region by writing it as an iterated integral
- use a double integral to calculate the area of a region, volume under a surface, or average value of a function over a plane region
additional notes
- the double integral as an iterated integral
  - set limits from the outside in
  - evaluate from the inside out
- partial integration
- constant limits and order of integration
  - fubini's theorem

recommended exercises
- 5, 8, 15, 17, 21, 25, 27, 35, 37, 45, 51, 53, 56, 57, 59, 63, 67
learning objectives
- recognize when a function of two variables is integrable over a general region
- evaluate a double integral by computing an iterated integral over a region bounded by two vertical lines and two functions of \( x \) or two horizontal lines and two functions of \( y \)
- simplify the calculation of an iterated integral by changing the order of integration
- use double integrals to calculate the volume of a region between two surfaces or the area of a plane region
- solve problems involving double improper integrals.
additional notes
- setting limits and evaluating multiple integrals
  - set limits from the "outside-in"
  - evaluate integrals from the "inside-out"
  - looking for one "entry curve" and one "exit curve"
- sketching regions of integration
  - the boundaries of the region translate directly into the limits of the integrals
- reversing the order of integration
  - may simplify a difficult (or impossible!) integration

Deadlines
- All homework is due at the beginning of class on the date indicated.
- All exams are given in class on the date indicated.
- Late homework is not accepted and there are no make-ups for missed exams. Contact me before the deadline if you have a conflict with a deadline and we will find a mutually agreeable solution.
- Solutions are posted under these links after the deadline.
Homework
- due Thursday, October 17:
  - 14.6 - , 14.7 - , 14.8 -
- due Thursday, October 24:
  - 15.1 - , 15.2 - , 15.3 -
- due Thursday, October 31:
  - 15.4 - , 15.5 - , 15.6 -
Exams
- Thursday, November 7: Exam 2
  - Exam 2 (solutions)
  - Exam 2 covers sections 14.4-14.8 and 15.1-15.7