1. Integration
Chapter 15

2. Integration 15.1 Antiderivatives 15.3 Area and Definite Integral
15.4 The Fundamental Theorem of Calculus

3. Differential and Integral Calculus
Differential calculus determines the rate of change at which a quantity is changing.
Example: f(x) = x5; find f ’(x).
Integral calculus finds the total change in quantity where the rate of change is known.
Example: f ’(x) = 5x4; find f(x).

4. 15.1 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 the derivative of F(x) = f(x), then F(x) is an antiderivative of f(x).
The process of finding antiderivatives is called antidifferentiation or indefinite integration.

5. 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)

6. Antiderivatives
Therefore, is an antiderivative of f(x) = x2 + 5

7. 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.

8. F(x), G(x), and H(x) are antiderivatives of 3x2
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. F(x), G(x), and H(x) are antiderivatives of 3x2
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.

10. 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

11. Antiderivatives ∫ 2ax dx = ∫ a(2x)dx = ax2 + C
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

12. 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.

14. 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)

15. 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 produced. Find the profit function.
(Used k instead of C to avoid confusion with “Cost”)

16. Rules For Finding Antiderivatives
INDEFINITE INTEGRALS OF EXPONENTIAL FUNCTIONS

17. Indefinite Integrals Of Exponential Functions
Examples

18. Rules For Finding Antiderivatives
POWER RULE FOR n = -1

19. Now You Try Find the cost function. Constant Multiple Rule Sum Rule
Power Rule

20. Now You Try
Find the cost function.
Constant Multiple Rule

21. 15.3 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]

22. f(x)
Area under the graph of x

23. 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.

24. f(x) x
Sum of the areas of the rectangles provides a rough approximation of the area under the curve.

25. f(x) x
Increasing the number of rectangles results in a closer approximation of the area under the curve.

26. f(x) x
Increasing the number of rectangles results in a closer approximation of the area under the curve.

27. 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 the 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)

28. f(x)
f(0) = 2 x 1

29. Area Under the Curve n
Area 2
3.732 4
3.496 8
3.340 10
3.305 20
3.229 50
3.178
100
3.160
500
3.146 n
Area
2000
3.143
8000
3.142
32,000
3.1417
128,000
3.1416
512,000

30. 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

31. 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

32. 15.4 The Fundamental Theorem of Calculus
As seen in section 15.3,
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 15.1
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)

33. 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.

34. y (100 ft)
Stream 1
x (100ft) 1

35. 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?

36. 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

37. 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

38. 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

39. Another example
Find the area below the graph of f(t) = 4t3 and above the x axis between t = 1 and t = 2.

40. Another example
Find the area below the graph of f(t) = 4t3 and above the x axis between t = 1 and t = 2.
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

42. Example (sum or difference of a function)
(constant multiple of a function)
(power rule)

43. Region lies below the x-axis
The area is given by

44. By the Fundamental Theorem,

45. Example
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?

46. Example
75 hours are required to produce the first 5 items.

47. y (hours) 75
x (items) 5
[0, 5]

48. Example
100 hours are required to produce the first 10 items. or

49. y (items)
x (hours)
[0, 10]

50. Now You Try.
Given: Find the area between the x axis and f(x) from x = 0 and x = 2.

51. 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.

52. Applications of Integrals
Consumers’ Surplus
Equilibrium Price
Producers’ Surplus

53. Applications of Integrals
Equilibrium Price

54. Applications of Integrals
Consumers’ Surplus
Equilibrium Price

55. Applications of Integrals
Consumers’ Surplus
If D(q) is a demand function with equilibrium price p0 and equilibrium quantity q0, then p0 q0

56. 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.

57. Applications of Integrals
Equilibrium Price

58. Applications of Integrals
Equilibrium Price
Producers’ Surplus

59. Applications of Integrals
Producers’ Surplus
If S(q) is a supply function with equilibrium price p0 and equilibrium quantity q0, then p0 q0

60. Applications of Integrals
Finding the consumers’ and producers’ surplus
Calculate equilibrium quantity (q0) and price (p0).

61. Applications of Integrals

62. Applications of Integrals

63. Applications of Integrals
$4500
p0 = $375
$3375
q0 = 15

64. Now You Try
Find the consumers’ and producers’ for an item having the following supply and demand functions

65. 
Chapter 15
End