Lessons 19, 20, and 21
Homework
19
20
21
Quizzes
Quiz 09
Quiz 10
Quiz 11
Quiz 12
Riemann Sums/Upper and Lower Sums/Definite Integrals
Indefinite integrals/Definite integrals/Applications/Avg value
Area/Consumer/Producer Surplus/Probability/Func of 2 Var.
Partial Deriv/Optimize functions of 2 Variables
Test 4
Lessons 16 – 24
Popper 19 and 20 in these notes!
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Lesson 19
Numerical Integration
For complicated formulas we will use GGB!
Example:
Evaluate 6 x 2  4e x 
1
x2
on the interval [1, 3]
Command: Integral [<function>, <start x>,<end x>]
Grab GGB
2
Evaluate
x2
1 3x3  6dx
2
Popper 19
Question 1
3
Applications of integrals
Example:
Consultants take data and fit a curve to a function that is the rate at which a worker
can produce items along an assembly line t hours after a worker’s shift has started.
N (t )  3t 2  12t  15
6
What is:
 N (t )dt ?
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The total number of items produced between 4 and 6 hours into the shift!
What is:
 N (t )dt
the total number of items produced
Find the answer to the first integral:
Example:
The marginal profit is the rate of change of the profit function. The antiderivative
of marginal profit is the total profit function. The area under the marginal profit
curve will give the total profit for given production levels.
A marginal profit function where x is the number of units produced and f (x) is
measured in dollars per unit is given:
f ( x)  0.0003x 2  0.04 x  17
Set up the math expression that represents the total increase in daily profits if the
company’s production goes from 200 units per day to 300 units per day:
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Find the dollar amount for this profit:
Popper 19 Question 2
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Example:
Velocity is the derivative of position when you are driving a car. If your velocity is
approximated by
v(t )  2t 25  t 2
Set up the expression that represents the distance traveled by the car in the first 5
seconds it is driven:
How far do you travel in those 5 seconds?
AVERAGE Value of a function
We will use the definite integral to find the average value of a function on the
interval [a, b]
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Here is the formula:
b
1
f ( x)dx
b  a a
You simply multiply the integral by the reciprocal of the length of the distance!
Example:
Find the average value of f ( x)  x2  3x  5 on [2. 5]
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Popper 19
Question 3
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Application:
The sales of a company in the first t years of existence is approximated by
S (t )  t 0.4t 2  4
Where S is in millions of dollars.
What is the company’s average income over the first 5 years of it’s existence?
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Lesson 20
Area Between Two Curves
The “general formula” for computing the area between two curves is
b
 top function  bottom function
dx
a
We’ve been using this with the x-axis as the bottom function…the formula for the
x-axis is y = 0…so you couldn’t really see the second part!
Example:
Find the area between the parabola f ( x)  9  x 2 and f ( x)  1 .
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Top function – bottom function is 9  x 2  1  8  x 2
Area = 30.17 sq units
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What does the “−1” do?
Why do these pictures tell a story?
Popper 19
Question 4
12
Example:
The graphs of the following three functions are shown. Find the area of the shaded
region (I’ll just add the shading!)
f ( x)   x  1
g ( x)  x  1
h( x )   x  1
Note that the bottom function CHANGES!
We’ll do ONE integral to zero and CHANGE functions for the next!
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Let’s write them out.
Get our LIMITS with INTERSECTION
Do the work!
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Popper 19
Question 5
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Example:
Find the area between
f ( x)  1.6 xe 0.23 x
on the interval [−0.07, 6.46]
g ( x)  0.5 x  3
Integral[f-g, -.07,6.46]
How to check: integrate f, integrate g…get the abs value and add!
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When the points of intersection are not given, use the command “Intersect”
Here’s an example:
f ( x)  1.3x3  2.8 x 2  8.1x  1.7
g ( x)  2.4 x 2  2 x  5
See the two area? Cubic – parabola on left, Parabola – cubic on right
Where are those points of intersection! GGB
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Note that I don’t care about the y-values, only the x-values.
What are the integrals we are going to do?
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Application Problem:
Two advertising agencies are competing for a major client. The rate of change of
the client’s revenue using Agency A’s campaign is approximated by f(x) below.
The rate of change for Agency B’s campaign is approximated by g(x) below.
If you graph these functions on the same grid…the area between the curves is the
additional revenue the client will realize by using the higher curve.
How would we find the additional revenue in the first 4 years of the proposed
campaigns?
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Example:
Without any effort to curb population growth, the government estimates that it’s
population will grow at the rate of
f ( x)  60e0.02 x
thousand people per year.
With an education program the growth rate will be altered to
g ( x)   x 2  60
thousand people per year over the next 5 years.
How many fewer people will there be in the country if the education program is
implemented and is successful?
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Popper 19
Question 6
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Example
The management of a hotel expects its profits to grow at a rate of
f (t )  1  t 2/3
million dollars per year t years from now.
If they renovate some of their existing hotels and acquire some new ones, their
profits would grow at the rate of
g (t )  t  2 t  4
million dollars per year
Tell how to find the additional profits the company could expect over the next 10
years if they proceed with their renovation and acquisition plans.
Enter the functions in GGB….which is the top function?
Set up the integral:
What are the commands, limits?
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End!
Popper 19 Questions 7 - 10
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Lesson 21
Other Applications of Integration
Consumer’s Surplus and Producer’s Surplus
The consumer’s surplus is defined to be the difference between what consumers
would be willing to pay and what they actually pay. It is the area of the region
bounded below by the demand function and above by the line that represents the
unit market price.
Then the consumers’ surplus is given by
CS 
QE
 D( x)dx  Q
E
 PE
0
In this formula, D(x) is the demand function, while
QE and PE are the equilibrium quantity and price.
The number subtracted is the equilibrium revenue!
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Notice that there is a second area shaded below the horizontal line through the
equilibrium point. Producers may be willing to sell their product for a lower price
than the prevailing market price. If the market price is higher than where producers
expect to price their items, then the difference is called the producers’ surplus.
The producer’s surplus is the area of the regionbounded below by the line that
represents the price and above by the supply curve.
It should come as no surprise that this, too, is and integral!
PS  QE  PE 
QE
 S ( x)dx
0
Again with equilibrium quantity and price, this time in the lead with the integral of
the supply curve on the interval [0, QE ].
Popper 20
Question 1
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Examples:
Suppose the demand for a certain product is given by
p D(x) 0.01x2 0.1x 6
where p is the unit price given in dollars and x is the quantity demanded per month
given in units of 1000. The market price for the product is $4 per unit.
Find the quantity demanded.
Enter the demand function into GGB. Then create a constant function f(x) = 4, in
order to find the point of intersection of the demand function and the constant
function. This answer will the quantity sold.
Find the consumers’ surplus if the market price for the product is $4 per unit.
Use
CS 
QE
 D( x)dx  Q
E
 PE
0
What will be the GGB commands? How will we tackle this?
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Popper 20
Question 2
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Sometimes, the unit price will not be given. Instead, product will be sold at market
price, and you’ll be given both supply and demand equations. In this case, we can
find the equilibrium point (remember: Section 4.2) which will give us the
equilibrium quantity and price.
Example:
The demand function for a popular brand of 12-speed bicycle is given by
p D(x) 0.001x2 250
where p is the unit price in dollars and x is the quantity demanded in
units of a thousand.
The supply function for the same product is given by
p S(x) 0.0006x2 0.02x 100
where p is the unit price in dollars and x is the quantity supplied in units of a
thousand.
Assume the market price is set at the equilibrium price. Find the equilibrium point.
Determine the consumers’ surplus.
CS 
QE
 D( x)dx  Q
E
 PE
0
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Determine the producers’ surplus.
PS  QE  PE 
QE
 S ( x)dx
0
Popper 20
Question 3
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Probability
Our last application area!
The study of probability deals with the likelihood of a certain outcome of an
experiment. When you flip a coin, the probability that the coin lands with heads
facing up is ½.
When you roll a six-sided die and record the number that lands on the uppermost
face, the probability that the die lands with the number 3 facing upwards is 1/6.
You can also find the probability of something occurring over a continuous interval.
Suppose you want to know how long a brand of light bulb lasts. Now the possible
set of answers contains more than a discrete set of numbers (e.g., something other
than 1, 2, 3, etc.). The light bulbs can last any positive length of time. We can find
the probability that the light bulbs life span is on a given interval [a, b].
This problem is an example of a problem that involves a probability density
function. This type of function can be used to determine the probability that an
event occurs on a given interval [a, b].
All probability density functions must meet these three criteria:
1.
2.
3.
f (x) 0 for all x
The area under the graph of f (x) is exactly 1.
The probability that an event occurs in an event [a, b] can be computed using
b
the definite integral:
 f ( x)dx .
a
Popper 20
Question 4
30
Example:
The function
f ( x)  0.02e0.002 x
gives the life span of a popular brand of light bulb, where x gives the lifespan in
hours and f (x) is the probability density function.
Find the probability that the lifespan is between 500 hours and 1000 hours.
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Example:
A company finds that the percent of its locations that experience a profit in the first
year of business has the probability density function
P( x) 
36
1
x(1  x) 2
11
3
0  x 1
100% THERE!
What is the probability that more than 50% of the company’s locations experienced
a profit during the first year of business?
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What is the probability that between 30% and 60% of the company’s locations
experienced a profit during the first year of business?
Popper 20
Question
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End Popper 20
Questions 6 – 10
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