1. Antiderivatives and Indefinite Integration
Lesson 5.1

2. Reversing Differentiation
An antiderivative of function f is
a ________________F
which satisfies __________
Consider the following:
We note that two antiderivatives of the same function differ by a __________________

3. Reversing Differentiation
General antiderivatives f(x) = 6x F(x) = 2x3 + C
because ___________ = 6x2
k(x) = sec2(x) K(x) = ___________________
because K’(x) = k(x)

4. Differential Equation
A differential equation in x and y involves
x, y, and _____________________ of y
Examples
Solution – find a function whose ___________is the differential given

5. Differential Equation
When
Then one such function is
The general solution is

6. Notation for Antiderivatives
We are starting with
Change to differential form
Then the notation for antiderivatives is
"The ______________of f with respect to x"

7. Basic Integration Rules
Note the inverse nature of integration and differentiation
Note basic rules, pg 286

8. Practice
Try these

9. Finding a Particular Solution
Given
Find the specific equation from the family of antiderivatives, which contains the point (3,2)
Hint: find the __________________, use the given point to find the value for C

10. Assignment A
Lesson 5.1 A
Page 291
Exercises 1 – 55 odd

11. Slope Fields Slope of a function f(x) Suppose we know f ‘(x) Example
at a point a
given by f ‘(a)
Suppose we know f ‘(x)
substitute different values for a
draw short slope lines for successive values of y
Example

12. Slope Fields For a large portion of the graph, when
We can trace the line for a specific F(x)
specifically when the C = -3

13. Finding an Antiderivative Using a Slope Field
Given
We can trace the version of the original F(x) which _______________________.

14. Vertical Motion
Consider the fact that the acceleration due to gravity a(t) = -32 fps
Then v(t) = -32t + v0
Also s(t) = -16t2 + v0t + s0
A balloon, rising vertically with velocity = 8 releases a sandbag at the instant it is 64 feet above the ground
How long until the sandbag hits the ground
What is its velocity when this happens?
Why?

15. Rectilinear Motion
A particle, initially at rest, moves along the x-axis at velocity of
At time t = 0, its position is x = 3
Find the velocity and position functions for the particle
Find all values of t for which the particle is at rest

16. Assignment B
Lesson 5.1 B
Page 292
Exercises 57 – 93, EOO

17. Area as the Limit of a Sum
Lesson 5.2

18. Area under f(x) = ln x
Consider the task to compute the area under a curve f(x) = ln x on interval [1,5] x
We estimate with 4 rectangles using the _________endpoints

19. Area under the Curve x
We can ________________our estimate by increasing the number of rectangles

20. Area under the Curve Increasing the number of rectangles to n
This can be done on the calculator:

21. Generalizing
a b
In general …
The actual area is
where

22. Summation Notation We use summation notation
Note the basic rules and formulas
Examples pg. 295
Theorem 5.2 Formulas, pg 296

23. Use of Calculator Note again summation capability of calculator
Syntax is:  (expression, variable, low, high)

24. Practice Summation
Try these

25. Practice Summation
For our general formula:
let f(x) = 3 – 2x on [0,1]

26. Assignment
Lesson 5.2
Page 303
Exercises 1 – 61 EOO (omit 45)

27. Riemann Sums and the Definite Integral
Lesson 5.3

28. Review We partition the interval into n ____________b
We partition the interval into n ____________
Evaluate f(x) at _________endpoints of kth sub-interval for k = 1, 2, 3, … n
f(x)

29. Review a b
Sum
We expect Sn to improve thus we define A, the ______________under the curve, to equal the above limit.
f(x)

30. Riemann Sum
Partition the interval [a,b] into n subintervals a = x0 < x1 … < xn-1< xn = b
Call this partition P
The kth subinterval is xk = xk-1 – xk
Largest xk is called the _________, called ||P||
Choose an arbitrary value from each subinterval, call it _________

31. Riemann Sum Form the sum This is the Riemann sum associated with
the function ______
the given partition ____
the chosen subinterval representatives ______
We will express a variety of quantities in terms of the Riemann sum

32. The Riemann Sum Calculated
Consider the function 2x2 – 7x + 5
Use x = 0.1
Let the = left edge of each subinterval
Note the sum

33. The Riemann Sum We have summed a series of boxes
If the x were ____________________, we would have gotten a better approximation
f(x) = 2x2 – 7x + 5

34. The Definite Integral
The definite integral is the _______of the Riemann sum
We say that f is _____________ when
the number I can be approximated as accurate as needed by making ||P|| sufficiently small
f must exist on [a,b] and the Riemann sum must exist

35. Example
Try
Use summation on calculator.

36. Example
Note increased accuracy with __________ x

37. Limit of the Riemann Sum
The definite integral is the ___________of the Riemann sum.

38. Properties of Definite Integral
Integral of a sum = sum of integrals
Factor out a _________________
Dominance

39. Properties of Definite Integral
Subdivision rule
f(x) a c b

40. Area As An Integral The area under the curve on the interval [a,b] A
f(x) A a c

41. Distance As An Integral
Given that v(t) = the velocity function with respect to time:
Then _____________________ can be determined by a definite integral
Think of a summation for many small time slices of distance

42. Assignment
Section 5.3
Page 314
Problems: 3 – 49 odd

43. The Fundamental Theorems of Calculus
Lesson 5.4

44. First Fundamental Theorem of Calculus
Given f is
_________________on interval [a, b]
F is any function that satisfies F’(x) = f(x)
Then

45. First Fundamental Theorem of Calculus
The definite integral can be computed by
finding an _________________F on interval [a,b]
evaluating at limits a and b and _____________
Try

46. Area Under a Curve
Consider
Area =

47. Area Under a Curve
Find the area under the following function on the interval [1, 4]

48. Second Fundamental Theorem of Calculus
Often useful to think of the following form
We can consider this to be a _______________ in terms of x
View QuickTime Movie

49. Second Fundamental Theorem of Calculus
Suppose we are given G(x)
What is G’(x)?

50. Second Fundamental Theorem of Calculus
Note that
Then
What about ?
Since this is a _____________

51. Second Fundamental Theorem of Calculus
Try this

52. Assignment
Lesson 5.4
Page 327
Exercises 1 – 49 odd

53. Integration by Substitution
Lesson 5.5

54. Substitution with Indefinite Integration
This is the “backwards” version of the _____________________
Recall …
Then …

55. Substitution with Indefinite Integration
In general we look at the f(x) and “split” it
into a ________________________
So that …

56. Substitution with Indefinite Integration
Note the parts of the integral from our example

57. Example Try this … We have a problem … what is the g(u)?
what is the du/dx?
We have a problem …
Where is the 4 which we need?

58. Example We can use one of the properties of integrals
We will insert a factor of _____inside and a factor of ¼ __________to balance the result

59. Can You Tell? Which one needs substitution for integration?
Go ahead and do the integration.

60. Try Another …

61. Assignment A
Lesson 5.5
Page 340
Problems: 1 – 33 EOO 49 – 77 EOO

62. Change of Variables
We completely rewrite the integral in terms of u and du
Example:
So u = _________ and du = _________
But we have an x in the integrand
So we solve for x in terms of u

63. Change of Variables We end up with
It remains to distribute the and proceed with the integration
Do not forget to "_________________"

64. What About Definite Integrals
Consider a variation of integral from previous slide
One option is to change the limits
u = __________ Then when t = 1, u = ___ when t = 2, u = ____
Resulting integral

65. What About Definite Integrals
Also possible to "un-substitute" and use the ___________________ limits

66. Integration of Even & Odd Functions
Recall that for an even function
The function is symmetric about the ________
Thus
An odd function has
The function is symmetric about the orgin

67. Assignment B
Lesson 5.5
Page 341
Problems: EOO 117 – 132 EOO

68. Numerical Integration
Lesson 5.6

69. Trapezoidal Rule
Instead of calculating approximation rectangles we will use trapezoids
More accuracy
Area of a trapezoid a b •
Which dimension is the h?
Which is the b1 and the b2 b1 b2 h

70. Trapezoidal Rule Trapezoidal rule approximates the integraldx
f(xi)
f(xi-1)
Trapezoidal rule approximates the integral
Calculator function for f(x) ((2*f(a+k*(b-a)/n),k,1,n-1)+f(a)+f(b))*(b-a)/(n*2) trap(a,b,n)

71. Trapezoidal Rule Entering the trapezoidal rule into the calculator
__________ must be defined for this to work

72. Trapezoidal Rule
Try using the trapezoidal rule
Check with integration

73. Simpson's Rule As before, we divide the interval into n parts
n must be ___________
Instead of straight lines we draw _____________through each group of three consecutive points
This approximates the original curve for finding definite integral – formula shown below a b •

74. Simpson's Rule Our calculator can do this for us also
The function is more than a one liner
We will use the program editor
Choose APPS, 7:Program Editor 3:New
Specify Function, name it simp

75. Simpson's Rule
Enter the parameters a, b, and n between the parentheses
Enter commands shown between Func and endFunc

76. Simpson's Rule Specify a function for ______________
When you call simp(a,b,n),
Make sure n is an number
Note the accuracy of the approximation

77. Assignment A
Lesson 5.6
Page 350
Exercises 1 – 23 odd

78. Error Estimation Trapezoidal error for f on [a, b]
Where M = _______________of |f ''(x)| on [a, b]
Simpson's error for f on [a, b]
Where K = max value of ___________ on [a, b]

79. Using Data
Given table of data, use trapezoidal rule to determine area under the curve
dx = ? x
2.00
2.10
2.20
2.30
2.40
2.50
2.60 y
4.32
4.57
5.14
5.78
6.84
6.62
6.51

80. Using Data
Given table of data, use Simpson's rule to determine area under the curve x
2.00
2.10
2.20
2.30
2.40
2.50
2.60 y
4.32
4.57
5.14
5.78
6.84
6.62
6.51

81. Assignment B
Lesson 5.6
Page 350
Exercises 27 – 39 odd 49, 51, 53

82. The Natural Log Function: Integration
Lesson 5.7

83. Log Rule for Integration
Because
Then we know that
And in general, when u is a differentiable function in x:

84. Try It Out
Consider these . . .

85. Finding Area
Given
Determine the area under the curve on the interval [2, 4]

86. Using Long Division Before Integrating
Use of the log rule is often in disguised form
Do the division on this integrand and alter it's appearance

87. Using Long Division Before Integrating
Calculator also can be used
Now take the integral

88. Change of Variables Consider So we have Then u = x – 1 and du = dx
But x = _________ and x – 2 = ______________
So we have
Finish the integration

89. Integrals of Trig Functions
Note the table of integrals, pg 357
Use these to do integrals involving trig functions

90. Assignment
Assignment 5.7
Page 358
Exercises 1 – 37 odd , 71, 73

91. Inverse Trigonometric Functions: Integration
Lesson 5.8

92. Review
Recall derivatives of inverse trig functions

93. Integrals Using Same Relationships
When given integral problems, look for these patterns

94. Identifying Patterns
For each of the integrals below, which inverse trig function is involved?

95. If they are not, how are they integrated?
Warning
Many integrals look like the inverse trig forms
Which of the following are of the inverse trig forms?
If they are not, how are they integrated?

96. Try These
Look for the pattern or how the expression can be manipulated into one of the patterns

97. Completing the Square
Often a good strategy when quadratic functions are involved in the integration
Remember … we seek _______________
Which might give us an integral resulting in the arctan function

98. Completing the Square
Try these

99. Rewriting as Sum of Two Quotients
The integral may not appear to fit basic integration formulas
May be possible to ______________________into two portions, each more easily handled

100. Basic Integration Rules
Note table of basic rules
Page 364
Most of these should be committed to memory
Note that to apply these, you must create the proper ________ to correspond to the u in the formula

101. Assignment
Lesson 5.8
Page 366
Exercises 1 – 39 odd
63, 67

102. Hyperbolic Functions -- Lesson 5.9
Consider the following definitions
Match the graphs with the definitions.
Note the identities, pg. 371

103. Derivatives of Hyperbolic Functions
Use definitions to determine the derivatives
Note the pattern or interesting results

104. Integrals of Hyperbolic Functions
This gives us antiderivatives (integrals) of these functions
Note other derivatives, integrals, pg. 371

105. Integrals Involving Inverse Hyperbolic Functions

106. Try It! Note the definite integral What is the a, the u, the du?
a = 3, u = _________, du = _______________

107. Application Find the area enclosed by x = -¼, x = ¼, y = 0, and
Which pattern does this match?
What is the a, the u, the du?

108. Assignment
Lesson 5.9
Page 377
Exercises 1 – 29 EOO – 53 EOO