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Integration
4.1
4.2
4.3
4.4
4.5
4.6
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Antiderivatives and Indefinite Integration
4.1
Objectives
Write the general solution of a differential equation.
Use indefinite integral notation for antiderivatives.
Use basic integration rules to find antiderivatives.
Find a particular solution of a differential equation.
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3. Antiderivatives The derivative of what is 2x? x2 x2 + 1 x2 – 5 ….
x2 + C
The derivative of what is sinθ
-cosθ
-cosθ + C
The derivative of what is 2? 2x
2x + 1
2x – 5 ….
2x + C
The derivative of what is 4x3 x4
x4 + C

4. Example 1 – Solving a Differential Equation
Find the general solution of the differential equation y' = 2.
y = 2x + C

5. Example 2 – Applying the Basic Integration Rules
Find the antiderivatives of 3x.
Find the integral of 2 sinx dx

6. Example 3

7. Example 4

8. Basic Integration Rules

9. Example 6

10. Example 7
Solve the differential equation with an initial condition F(2) = 4

11. Example 7 – Finding a Particular Solution
Find the general solution of and find the
particular solution that satisfies the initial condition F(1) = 0.

12. Example 7 – Solution So, the particular solution
cont’d
So, the particular solution
in that satisfies F(1) = 0 is

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4.1 Summary
Objectives
Write the general solution of a differential equation.
Use indefinite integral notation for antiderivatives.
Use basic integration rules to find antiderivatives.
Find a particular solution of a differential equation.
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14. 4.2 Area and Summations Sigma Notation Summation Formulas
Approximating the Area of a Plane Region
Finding the Area by the Limit Definition

15. Ex 1 Sigma Notation ≈
i =
k =

16. Ex 2 Use sigma notation to write the sum

17. Ex 3 Use the summation formulas to evaluate

18. Ex 3 Use the summation formulas to evaluate

19. Ex 4 Use summation formulas to rewrite the expression without the summation notation

20. Ex 4 Use summation formulas to rewrite the expression without the summation notation

21. Ex 6 Use the summation formulas and find the limit

22. Ex 6 Use the summation formulas and find the limit

23. Ex 6 Use the summation formulas and find the limit

24. Ex 5 Use left and right endpoints to estimate the area
f(x) = 2x + 3, [0,2], 4 rectangles
Draw the picture
Make an x,y table
Calculate the area X Y 1
3/2 2 3 4 5 6
Δx = (2 – 0)/4 = 0.5 7
Area (rect left endpts) = 0.5( ) = 9
Area (rect rt endpts) = 0.5( ) = 11

25. Ex 5 Use left and right endpoints to estimate the area
[0,1], 4 rectangles
Draw the picture
Make an x,y table
Calculate the area X Y 1
√3/4
√3/2
√9/4
Δx = (1 – 0)/4 = 0.25 √3
Area (rect left endpts) = 0.25(0 + √3/4 + √3/2 + √9/4) =
Area (rect rt endpts) = 0.25(√3/4 + √3/2 + √9/4 + √3) =

26. 4.2 Summary Sigma Notation Summation Formulas
Approximating the Area of a Plane Region
Finding the Area by the Limit Definition

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4.3
Definite Integrals
Objectives
Find the area under a curve using familiar geometric shapes.
Evaluate a definite integral.
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28. Example 1 – Evaluating a Definite Integral as a Limit
Evaluate the definite integral
The function f(x) = 2x is integrable on the interval [–2, 1] because it is continuous on [–2, 1].

29. Definite Integrals

30. Example 2
Find the area of the region bounded by the graph of f(x) = 4x – x2 and the x-axis
Because f is continuous and nonnegative
on the closed interval [0, 4], the area of
the region is
Also show the buttons on the calculator math and graph…

31. Example 3 – Areas of Common Geometric Figures
Sketch the region corresponding to each definite integral. Then evaluate each integral using a geometric formula.
a b c.
½ circ = ½ π(2)2
Rect = 2x4
Trap = ½ (3)(2+5)

32. Properties of Definite Integrals

33. Example 4 – Evaluating Definite Integrals
a. Find the value of
Given
find the value of

34. Example 5 – Using the Additive Interval Property-1 1

35. Example 6 – Evaluation of a Definite Integral
Evaluate.

36. Write an integral to represent the area
f(x) = 25 − x2
g(y) = y3

37. = -1/2 = -1/2 + 2 = 3/2 = -5(2 – 4) = -3/2 = 10 = 4(2 + 2 – 4 + ½)
The graph of f consists of line segments, as shown in the figure. Evaluate each definite integral by using geometric formulas.
= -1/2
= -1/2 + 2
= 3/2
= -5(2 – 4)
= -3/2
= 10
= 4(2 + 2 – 4 + ½)
= -21/2
= 2

38. Area
Find the area between the curve and the x-axis from x = -2 to x = 4 for f(x) = x2 – 1

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4.3 Summary
Objectives
Find the area under a curve using familiar geometric shapes.
Evaluate a definite integral.
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4.4
The Fundamental Theorem of Calculus
Objectives
Evaluate a definite integral using the Fundamental Theorem of Calculus.
Understand and use the Mean Value Theorem for Integrals.
Find the average value of a function over a closed interval.
Understand and use the Second Fundamental Theorem of Calculus.
Understand and use the Net Change Theorem.
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41. The Fundamental Theorem of Calculus
The two major branches of calculus: differential calculus and integral calculus. At this point, these two problems might seem unrelated—but there is a very close connection.
The connection was discovered independently by Isaac Newton and Gottfried Leibniz and is stated in a theorem that is appropriately called the Fundamental Theorem of Calculus.

42. The Fundamental Theorem of Calculus

43. Example 1
Evaluate each definite integral.

44. Example 2
Evaluate 2

45. Example 3
Find the area of the region bounded by the graph of y = 2x2 – 3x + 2, the x-axis, and the vertical lines x = 0 and x = 2.
Vertex: (-b/2a, )
(3/4, 7/8) opens up 2

46. The Mean Value Theorem for Integrals
The area of a region under a curve is greater than the area of an inscribed rectangle and less than the area of a circumscribed rectangle. The Mean Value Theorem for Integrals states that somewhere “between” the inscribed and circumscribed rectangles there is a rectangle whose area is precisely equal to the area of the region under the curve.

47. The Mean Value Theorem for Integrals
The value of f(c) given in the Mean Value Theorem for Integrals is called the average value of f on the interval [a, b].

48. Average Value of a Function
In Figure 4.31 the area of the region under the graph of f is equal to the area of the rectangle whose height is the average value.
Figure 4.31

49. Example 4 – Finding the Average Value of a Function
Find the average value of f(x) = 3x2 – 2x on the interval [1, 4].
The average value is given by

50. Example 6 – The Definite Integral as a Function
Evaluate the function
You could evaluate five different definite integrals, one for each of the given upper limits.
However, it is much simpler to fix x (as a constant) temporarily to obtain

51. The Second Fundamental Theorem of Calculus

52. The Second Fundamental Theorem of Calculus

53. Example 7 – Using the Second Fundamental Theorem of Calculus
Evaluate
Note that is continuous on the entire real line.
So, using the Second Fundamental Theorem of Calculus, you can write

54. Example 8
Find the derivative of each:

55. Net Change Theorem

56. Example 9 – Using the Net Change Theorem
A chemical flows into a storage tank at a rate of t liters per minute, where 0 ≤ t ≤ 60. Find the amount of the chemical that flows into the tank during the first 20 minutes.
Let c(t) be the amount of the chemical in the tank at time t.
Then c'(t) represents the rate at which the chemical flows into the tank at time t.
So, the amount that flows into the tank during the first 20 minutes is 4200 liters.

57. Net Change Theorem
The velocity of a particle moving along a straight line where s(t) is the position at time t. Then its velocity is v(t) = s'(t) and
This definite integral represents the net change in position, or displacement, of the particle.

58. Net Change Theorem
When calculating the total distance traveled by the particle, you must consider the intervals where v(t) ≤ 0 and the intervals where v(t) ≥ 0.
When v(t) ≤ 0 the particle moves to the left, and when v(t) ≥ 0, the particle moves to the right.
To calculate the total distance traveled, integrate the absolute value of velocity |v(t)|.

59. Net Change Theorem
So, the displacement of a particle and the total distance traveled by a particle over [a, b] can be written as

60. Example 10 – Solving a Particle Motion Problem
A particle is moving along a line so that its velocity is v(t) = t3 – 10t2 + 29t – 20 feet per second at time t.
What is the displacement of the particle on the time
interval 1 ≤ t ≤ 5?
b. What is the total distance traveled by the particle on the
time interval 1 ≤ t ≤ 5?

61. Example 10(a) – Solution
By definition, you know that the displacement is
So, the particle moves feet to the right.

62. Ex 10(b) – Solution To find the total distance traveled, calculate
v(t) = t3 – 10t2 + 29t – 20
cont’d
To find the total distance traveled, calculate
v(t) = (t – 1)(t – 4)(t – 5), you can determine that v(t) ≥ 0 on [1, 4] and on v(t) ≤ 0 on [4, 5].

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4.4 Summary
Objectives
Evaluate a definite integral using the Fundamental Theorem of Calculus.
Understand and use the Mean Value Theorem for Integrals.
Find the average value of a function over a closed interval.
Understand and use the Second Fundamental Theorem of Calculus.
Understand and use the Net Change Theorem.
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65. 4.5 Integration by Substitution Objectives
Use a change of variables to find an indefinite integral.
Use the General Power Rule for Integration to find an indefinite integral.
Use a change of variables to evaluate a definite integral.
Evaluate a definite integral involving an even or odd function.
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66. Pattern Recognition
In this section you will study techniques for integrating composite functions. You will be using a technique with u-substitution.
The role of substitution in integration is comparable to the role of the Chain Rule in differentiation.

67. Example 1 & 2
Find
u = x2 + 1
du = 2x dx
Find

68. Example 3 – Multiplying and Dividing by a Constant
Find

69. Example 4
Find

70. Example 5
Find

71. Example 6
Find

72. Example 7 – Substitution and the General Power Rule

73. Example 7 – Substitution and the General Power Rule
cont’d

74. Change of Variables for Definite Integrals
When using u-substitution with a definite integral, it is often convenient to determine the limits of integration for the variable u rather than to convert the antiderivative back to the variable x and evaluate at the original limits.
This change of variables is stated explicitly in the next theorem.

75. Example 8 – Change of Variables
Evaluate
Before substituting, determine the new upper and lower limits of integration.

76. Ex: 9

77. Integration of Even and Odd Functions
Even with a change of variables, integration can be difficult.
Occasionally, you can simplify the evaluation of a definite integral over an interval that is symmetric about the y-axis or about the origin by recognizing the integrand to be an even or odd function

78. Integration of Even and Odd Functions

79. Example 10 – Integration of an Odd Function
Evaluate
f(x) = sin3x cos x + sin x cos x
f(–x) = sin3(–x) cos (–x) + sin (–x) cos (–x)
= –sin3x cos x – sin x cos x = –f(x)
Which means that f(x) is odd …

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4.5 Summary
Objectives
Use a change of variables to find an indefinite integral.
Use the General Power Rule for Integration to find an indefinite integral.
Use a change of variables to evaluate a definite integral.
Evaluate a definite integral involving an even or odd function.
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81. 4.6 Numerical Integration Area under a curve by estimation methods:
Left Rectangles
Right Rectangles
Trapezoids
Simpsons Rule
Area under a curve EXACT
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82. Ex 1 Estimate w/ left & right rectangles, trap, and simpsons rule
Ex 1 Estimate w/ left & right rectangles, trap, and simpsons rule. Then find exact answer
f(x) = 2x + 3, [0,2], 4 rectangles
Draw the picture
Make an x,y table
Δx = (2 – 0)/4 = 0.5 X Y 1
3/2 2
Calculate the area 3 4 5 6 7
Area (rect left endpts) = 0.5( ) = 9
Area (rect rt endpts) = 0.5( ) = 11
Area (trapezoids) = ½ * 0.5(3 + 2*4 + 2*5 + 2*6 + 7) = 10
Area (Simpsons) = 1/3 * 0.5(3 + 4*4 + 2*5 + 4*6 + 7) = 10

83. Ex 2 Estimate w/ left & right rectangles, trap, and simpsons rule
Ex 2 Estimate w/ left & right rectangles, trap, and simpsons rule. Then find exact answer
[0,1], 4 rectangles
Δx = (1 – 0)/4 = 0.25 X Y 1
√3/4
√3/2
√9/4 √3
Area (rect rt endpts) = 0.25(√3/4 + √3/2 + √9/4 + √3) =
Area (rect left endpts) = 0.25(0 + √3/4 + √3/2 + √9/4) =
Area (trapezoids) = 0.5*0.25(0 + 2*√3/4 + 2*√3/2 + 2*√9/4 + √3) =
Area (Simpson’s) = (1/3)*0.25(0 + 4*√3/4 + 2*√3/2 + 4*√9/4 + √3) =

84. On the test you will be given:

86. The Trapezoidal Rule
One way to approximate a definite integral is to use n trapezoids, as shown in Figure 4.42.
In the development of this method,
assume that f is continuous and
positive on the interval [a, b].
So, the definite integral
represents the area of the region
bounded by the graph of f and the
x-axis, from x = a to x = b.
Figure 4.42

87. The Trapezoidal Rule The area of the ith trapezoid is
This implies that the sum of the areas of the n trapezoids

88. The Trapezoidal Rule Letting you can take the limits as to obtain
The result is summarized in the following theorem.

89. The Trapezoidal Rule

90. Example 1 – Approximation with the Trapezoidal Rule
Use the Trapezoidal Rule to approximate
Compare the results for n = 4 and n = 8, as shown in Figure 4.44.
Figure 4.44

91. Example 1 – Solution
When n = 4, ∆x = π/4, and you obtain

92. Example 1 – Solution
cont’d
When and you obtain

93. Simpson’s Rule
One way to view the trapezoidal approximation of a definite integral is to say that on each subinterval you approximate f by a first-degree polynomial.
In Simpson’s Rule, named after the English mathematician Thomas Simpson (1710–1761), you take this procedure one step further and approximate f by second-degree polynomials.
Before presenting Simpson’s Rule, we list a theorem for evaluating integrals of polynomials of degree 2 (or less).

94. Simpson’s Rule

95. Simpson’s Rule
To develop Simpson’s Rule for approximating a definite integral, you again partition the interval [a, b] into n subintervals, each of width ∆x = (b – a)/n.
This time, however, n required to be even, and the subintervals are grouped in pairs such that
On each (double) subinterval [xi – 2, xi ], you can approximate f by a polynomial p of degree less than or equal to 2.

96. Simpson’s Rule
For example, on the subinterval [x0, x2], choose the polynomial of least degree passing through the points (x0, y0), (x1, y1), and (x2, y2), as shown in Figure 4.45.
Figure 4.45

97. Simpson’s Rule
Now, using p as an approximation of f on this subinterval, you have, by Theorem 4.18,
Repeating this procedure on the entire interval [a, b]
produces the following theorem.

98. Simpson’s Rule

99. Example 2 – Solution
Use Simpson’s Rule to approximate for n = 4 and n = 8
When n = 4, you have
when n = 8, you have

100. Error Analysis
If you must use an approximation technique, it is important to know how accurate you can expect the approximation to be.
The following theorem, gives the formulas for estimating the errors involved in the use of Simpson’s Rule and the Trapezoidal Rule.
In general, when using an approximation, you can think of the error E as the difference between and the approximation.

101. Error Analysis

102. Example 3 – The Approximate Error in the Trapezoidal Rule
Determine a value of n such that the Trapezoidal Rule will approximate the value of with an error that is less than or equal to 0.01.
Solution:
Begin by letting and finding the second derivative of f.
The maximum value of |f''(x)| on the interval [0, 1] is
|f''(0)| = 1.

103. Example 3 – Solution So, by Theorem 4.20, you can write
cont’d
So, by Theorem 4.20, you can write
To obtain an error E that is less than 0.01, you must choose n such that 1/(12n2) ≤ 1/100.

104. Example 3 – Solution
cont’d
So, you can choose n = 3 (because n must be greater than or equal to 2.89) and apply the Trapezoidal Rule, as shown in Figure 4.46, to obtain
Figure 4.46

105. Example 3 – Solution
cont’d
So, by adding and subtracting the error from this estimate, you know that