9 FURTHER APPLICATIONS OF INTEGRATION

9.4 APPLICATIONS TO ECONOMICS AND BIOLOGY
SUGGESTED TIME AND EMPHASIS
Sections 9.2–9.5 are not covered directly on the Calculus AB exam. However, a major part of the study of
calculus is applications. Something would be lost if students left a calculus course without a good idea of the
variety of ways in which their knowledge could be applied. It is therefore recommended that some time be
spent on a subset of these sections, either immediately after covering Section 9.1, or during the interval of
time between the AP exam and the end of the course.
POINTS TO STRESS
Three applications (consumer surplus, blood flow, and cardiac output) are discussed in the text. It is recommended
that the instructor choose either consumer surplus or blood flow to stress in lecture, and perhaps touch
on the other two examples in workshop.
QUIZ QUESTIONS
• Text Question: What is the meaning of the term “flux”?
Answer: In the context of blood flow, it refers to the volume per unit time of blood that passes through a
cross-section of a blood vessel.
• Drill Question: The marginal cost of producing x cases of olive oil is 74+1.1x−0.002x 2 +0.00004x 3 (in
dollars per case). Find the increase in cost if the production level is raised from 1200 cases to 1600 cases.
Answer: $43,866,933.33
MATERIALS FOR LECTURE AND WORKSHOP/DISCUSSION
• Provide a complete description of one of the three examples in the text. Touch on the other two examples.
• Surplus: Bring a quantity of dice, stickers, or any other inexpensive product to class, one for every student.
Secretly determine your selling price P. Have the students write down the most that they would pay for
one of the objects. (In a smaller class, give a definite range of prices.) For example, if Brandon’s price
is lower than your secret price P, no transaction will take place. If his price is higher than P, hewillbuy
the object. Daniel’s individual savings is the difference between the price he was willing to pay and P. In
the example below, his savings would be 50 − 35 = 15 cents. The consumer surplus for the class will be
the sum of these savings. Construct a demand curve p (x) by tallying their bids in decreasing order and
computing the consumer surplus numerically using a step function. Discuss why we would model p (x)
as a continuous function in some cases.
Example: Suppose you bring in your collection of McGovern/Eagleton buttons, and you are willing to part
with them for 35 cents each. Assume your students wrote down the most they would pay for one:
Michael 0.15 Daniel 0.50 Christopher 1.50 Jessica 0.25 Ashley 0.35
Joshua 0.10 Samantha 0.80 Amanda 0.40 Brittany 0.60 Matthew 1.25
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