Integration

1
Integration

• Chapter 13

2
Integration

• 13.1 Antiderivatives
• 13.3 Area and Definite Integral
• 13.4 The Fundamental Theorem of Calculus
• 13.5 Applications of Integrals

3
Differential and Integral Calculus

• Differential calculus determines the rate of
  change of a quantity.
• The derivative is related to the slope of the
  tangent line to a curve.
• Integral calculus finds the quantity where the
  rate of change is known.
• The definite integral is related to the area
  under a curve.

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13.1 Antiderivatives

• In previous chapters, functions provided
  information about a total amount q.
• Cost C(q), revenue R(q), profit P(q),...
• Derivatives of these functions provide
  information about the rate of change and extrema.
• If the rate of change is known, the
  antiderivative is calculated to determine the
  function and the total quantity.

ANTIDERIVATIVE If F(x) f(x), then F(x) is an
antiderivative of f(x) The process of finding
antiderivatives is called antidifferentiation or
indefinite integration.
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Antiderivatives
Examples

• If F(x) 10x, then F(x) 10, so F(x) 10x is
  an antiderivative of f(x) 10.
• For F(x) x2, F(x) 2x, making F(x) x2 an
  antiderivative of f(x) 2x.
• Find an antiderivative of f(x) 5x4
• f(x) F(x) 5x5 1
• Therefore, the antiderivative of f(x) x5
• Verify that is
  an antiderivative of f(x) x2 5
• F(x) is an antiderivative of f(x) if, and only
  if, F(x) f(x)

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Antiderivatives

• Therefore, is an
  antiderivative of f(x) x2 5

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The General Antiderivative of a Function

• A function has more than one antiderivative.
• One antiderivative of the function f(x) 3x2 is
  F(x) x3, since
• F(x) 3x2 f(x)
• but so are x3 10, and x3 5, and x3 ?, since
• This means that there is a family or class of
  functions that are antiderivatives of 3x2.

If F(x) and G(x) are both antiderivatives of a
function f(x) on an interval, than there is a
constant C such that F(x) G(x) C Two
antiderivatives of a function can differ only by
a constant. The arbitrary real number C is called
an integration constant.
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G(x) g(x) 3x2
F(x) f(x) 3x2
H(x) h(x) 3x2
F(x), G(x), and H(x) are antiderivatives of 3x2
9
C 10
F(x), G(x), and H(x) are antiderivatives of 3x2
C -5
Each graph can be obtained from another by a
vertical shift of C units. The family of
antiderivatives of f(x) is represented by
F(x) C.
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Antiderivatives

• The family of all antiderivatives of the function
  f is indicated by
• The symbol ? is the integral sign, f(x) is the
  integrand, and ? f(x) dx is called an indefinite
  integral, the most general antiderivative of f.

INDEFINITE INTEGRAL If F(x) f(x), then for
any real number C
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Antiderivatives

• Using this notation,
• ? 2x dx x2 C
• the dx indicates that ? f(x) dx is the integral
  of f(x) with respect to x.
• In ? 2ax dx, the dx indicates that a is a
  constant and x is the variable of the function.
  Therefore,

• ? 2ax dx ? a(2x)dx ax2 C
• And
• ? 2ax da ? x(2a) da xa2 C

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Rules For Finding Antiderivatives

• CONSTANT MULTIPLE RULE
• For constant k
• The indefinite integral of a constant is the
  constant itself.

13
Rules For Finding Antiderivatives

• POWER RULE
• For any real number n ? -1,
• To find the indefinite integral of a variable x
  raised to an exponent n, increase the exponent by
  1 and divide by the new exponent, n 1.

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Rules For Finding Antiderivatives

• SUM OR DIFFERENCE RULE
• The indefinite integral of the sum or difference
  of two functions f(x) and g(x) with respect to x,
  is the sum or difference of the indefinite
  integrals of the functions.

(Constant rule)
(Power rule)
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Example

• The marginal profit of a small fast-food stand is
  given by
• P(x) 2x 20
• where x is the sales volume in thousands of
  hamburgers. The profit is 50 when no
  hamburgers are sold. Find the profit function.

(Used k instead of C to avoid confusion with
Cost)
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Rules For Finding Antiderivatives

• INDEFINITE INTEGRALS OF EXPONENTIAL FUNCTIONS

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Indefinite Integrals Of Exponential Functions

• Examples

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Rules For Finding Antiderivatives

• POWER RULE FOR n -1

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Now You Try

• Find the cost function.

20
Now You Try

• Find the cost function.

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13.3 Area and the Definite Integral

• Definite integral Can be defined as the area of
  the region under the graph of a function f on the
  interval a, b

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f(x)
Area under the graph of
x
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Area and the Definite Integral

• Definite integral Can be defined as the area of
  the region under the graph of a function f on the
  interval a, b
• Based upon the concept that a geometric figure is
  a sum of other figures.

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f(x)
x
Sum of the areas of the rectangles provides a
rough approximation of the area under the curve.
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f(x)
x
Increasing the number of rectangles results in a
closer approximation of the area under the curve.
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f(x)
x
Increasing the number of rectangles results in a
closer approximation of the area under the curve.
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Area and the Definite Integral

• Definite integral Can be defined as the area of
  the region under the graph of a function f on the
  interval a, b
• Based upon the concept that geometric figure is a
  sum of other figures.
• The area under the curve can be determined by
  summing the areas of n rectangles of equal width.
• If the interval is from x 0 to x 2, then the
  width of each rectangle equals
• and height determined by f(x)

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f(x)
f(0) 2
x
1
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Area Under the Curve
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Definite Integral

• Calculating the area under the curve of a
  function on the interval a, b
• Divide the area between x a, and x b into n
  intervals
• Let
• f(xi) the height if the ith rectangle
• (The ith rectangle is an arbitrary rectangle)
• ?x the width of each of the rectangles
• Then the area of the ith rectangle f(xi) ?x
• and the total area under the curve is
  approximated by the sum of the areas of all n
  rectangles, or

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Now You Try.
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Definite Integral

• THE DEFINITE INTEGRAL
• If f is defined by the interval a, b, the
  definite integral of f from a to b is given by
• provided the limit exists, where ?x (b
  a)/n and xi is any value of x in the ith interval.

The definite integral of f(x) on a, b is
written as
and approximated by
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13.4 The Fundamental Theorem of Calculus

• As seen in section 13.3,
• gives the area between the graph of f(x) and the
  x-axis, from x a to x b.
• This area can be determined using the
  antiderivatives discussed in section 13.1
• If f(x) gives the rate of change of F(x), then
  F(x) is an antiderivative of f(x)
• and gives the total change of F(x)
  as x changes from a to b
• which can also be expressed as F(b) F(a)

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The Fundamental Theorem of Calculus

• A real estate agent wants to evaluate an
  unimproved parcel of land that is 100 feet wide
  and is bounded by streets on three sides and by a
  stream on the fourth side. The agent determines
  that if a coordinate system is set up as shown in
  the next slide, the stream can be described by
  the curve y x3 1, where x and y are measured
  in hundreds of feet.

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y (100 ft)
Stream
1
x (100ft)
1
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The Fundamental Theorem of Calculus

• A real estate agent wants to evaluate an
  unimproved parcel of land that is 100 feet wide
  and is bounded by streets on three sides and by a
  stream on the fourth side. The agent determines
  that if a coordinate system is set up as shown in
  the next slide, the stream can be described by
  the curve y x3 1, where x and y are measured
  in hundreds of feet. If the area of the parcel is
  A square feet and the agent estimates the land is
  worth 12 per square foot, then the total value
  of the parcel is 12A dollars. How can the agent
  find the area and hence the total value of the
  parcel?

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The Fundamental Theorem of Calculus

• Allows us to compute definite integrals and thus
  find area and other quantities by using the
  indefinite integration methods.

THE FUNDAMENTAL THEOREM OF CALCULUS If the
function f(x) is continuous on the interval a,
b, then where F(x) is any antiderivative of
f(x) on a, b
When applying the Fundamental Theorem, use the
notation
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The Fundamental Theorem of Calculus

• Find the area of the parcel of land described in
  the introduction to this section. That is, the
  area under the curve y x3 1 on the interval
  0, 1

The area of the parcel is given by the definite
integral
Since an antiderivative of f(x) x3 1 is
the fundamental theorem of calculus tells us that
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The Fundamental Theorem of Calculus
Because x and y are measured in hundreds of feet,
the total area is
Since the land is worth 12 per square foot, the
total value of the parcel is
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Another example
Solution
41
Properties of Definite Integrals
for any real constant k
(constant multiple of a function)
(sum or difference of a function)
for any real number c that lies between a and b
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Example
(sum or difference of a function)
(constant multiple of a function)
(power rule)
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Region lies below the x-axis
The area is given by
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By the Fundamental Theorem,
45
Pg. 838, 52

• A worker new to a job will improve his efficiency
  with time so that it takes him fewer hours to
  produce an item with each day on the job, up to a
  certain point.

46
Pg. 838, 52

• A worker new to a job will improve his efficiency
  with time so that it takes him fewer hours to
  produce an item with each day on the job, up to a
  certain point.
• Suppose the rate of change of the number of hours
  it takes a worker to produce the xth item is
  given by
• H(x) 20 2x

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Pg. 838, 52

• A worker new to a job will improve his efficiency
  with time so that it takes him fewer hours to
  produce an item with each day on the job, up to a
  certain point.
• Suppose the rate of change of the number of hours
  it takes a worker to produce the xth item is
  given by
• H(x) 20 2x
• What is the total number of hours required to
  produce the first 5 items?
• What is the total number of hours required to
  produce the first 10 items?

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Pg. 838, 52
75 hours are required to produce the first 5
items.
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y (hours)
75
x (items)
5
0, 5
50
Pg. 838, 52
100 hours are required to produce the first 10
items.
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y (items)
x (hours)
0, 10
52
Now You Try.
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13.5 Applications of Integrals

• A car-leasing firm must decide how much to charge
  for maintenance on the cars it leases. After
  careful study, the firm decides that the rate of
  maintenance, m(x), on a new car will approximate
• m(x) 60(1 x2),
• where x is the number of years the car has been
  in use.
• What total maintenance cost can the company
  expect for a 2-year lease?
• What minimum amount should be added to the
  monthly lease payments to pay for maintenance?

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Applications of Integrals
The total maintenance charge on a 2-year lease is
given by
Monthly addition for maintenance
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Applications of Integrals

• Using the previous exercise, find the maintenance
  cost the company can expect during the third year.

The maintenance charge for the 3rd year is 440.
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Applications of Integrals

• Consumers and Producers Surplus
• Equilibrium price The price at which quantity
  demanded equals quantity supplied.
• Some buyers are willing to pay more.
• Consumers Surplus The total of the differences
  between the equilibrium price and the higher
  prices consumers would be willing to pay.

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Applications of Integrals
Consumers Surplus
Equilibrium Price
Producers Surplus
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Applications of Integrals
Equilibrium Price
Producers Surplus
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Applications of Integrals
Consumers Surplus
Equilibrium Price
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Applications of Integrals

• Consumers Surplus
• If D(q) is a demand function with equilibrium
  price p0 and equilibrium quantity q0, then

p0
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Applications of Integrals

• Consumers and Producers Surplus
• Equilibrium price The price at which quantity
  demanded equals quantity supplied.
• Some buyers are willing to pay more.
• Consumers Surplus The total of the differences
  between the equilibrium price and the higher
  prices consumers would be willing to pay.
• Producers Surplus The total of the differences
  between the equilibrium price and the lower
  prices sellers would be willing to charge.

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Applications of Integrals
Equilibrium Price
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Applications of Integrals
Equilibrium Price
Producers Surplus
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Applications of Integrals

• Producers Surplus
• If S(q) is a supply function with equilibrium
  price p0 and equilibrium quantity q0, then

p0
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Applications of Integrals

• Finding the consumers and producers surplus
• Calculate equilibrium quantity (q0) and price
  (p0).

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Applications of Integrals
67
Applications of Integrals
68
Applications of Integrals
4500
p0 375
3375
q0 15
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Now You Try, pg. 850, 24
70
?
Chapter 13
End