Math 31L.13
Calculus I with Environmental Emphasis

Fall 1997

Plan for Weeks 12 and 13

Most of this course has been devoted to one of the two major branches of calculus, differential calculus. To provide a glimpse of "the other side of the story" -- and to prepare for Calculus II -- we now devote five lessons and one lab to the other major branch, integral calculus.

The key operations in differential calculus are subtraction and division; for integral calculus, the key operations are multiplication and addition. In Week 12 we encounter the integral, first as a measure of accumulation when a rate of change is known. For example, if we know the speed of a moving car at every time during a trip, we can calculate the distance traveled during the trip. That is, the odometer "integrates" the speedometer. We will use the same idea in the lab to calculate a total air pollution burden from data that gives a pollution rate.

At the end of Week 12 we consider the integral as an “averaging” or “smoothing” operation. In particular, we pass from the familiar idea of averaging a list of numbers to the closely related idea of averaging a continuous function.

In our shortened Week 13, we establish a connection -- the Fundamental Theorem of Calculus -- between the concepts of integral and derivative. This connection gives added importance to the idea of antiderivative, which we have used often in solving differential equations.

The process of antidifferentiation is of course the inverse of the process of differentiation. Our preparatory work shows it is no accident that this inverse process is closely related to integration. First, as noted already, the key operations in integration are multiplication and addition -- inverses of the processes of division and subtraction. Second, we can think of the limiting process in the derivative as moving from descriptions of average rate of change over intervals (difference quotients) to instantaneous rate of change. The integral (seen as an averageing operation) reverses that process by taking us from instantaneous rate of change to average rate of change over an interval.

Here is the syllabus for Weeks 12 and 13:

Week 12 Date Topic Reading Activity
M 11/17 Distance from velocity 3.1 Start Take-Home Test 2
W 11/19 Definite integral 3.2 Take-Home Test 2 due
Th 11/20 Air pollution Lab: Accumulation
F 11/21 Area and averaging 3.3
Week 13 Date Topic Reading Activity
M 11/24 Fundamental Theorem 3.4
W 11/26 Fundamental Theorem 3.4
Thanksgiving Break

Notes

  1. You are expected to read the relevant section of the text before the class period using it.
  2. Your homework papers for Week 12 will be turned in on Monday, November 24. The papers for Week 13 will be turned in on Monday, December 1. Homework papers should include solutions to the problems whose numbers are boldfaced italics in the assignments below. The assignment dates are start dates.
  3. It is strongly recommended that you do the Week 13 homework assignment before Thanksgiving break. There are two class periods on Section 3.4, but all of the problems will be accessible after the Monday class. This assignment is not as long as it looks -- each of the problem groups 15-18 and 19-21 is really a multiple part problem.
  4. Remember to submit your Week 12 journal entry by e-mail on or before Friday, November 21. No journal entry will be required in Week 13.

Assignments

David A. Smith <das@math.duke.edu>

Last modified: November 15, 1997