1. Integration Antidfferentiation
The Fundamental Theorem of Calculus (FTC)
and the Algebra of Integration
Second FTC

2. Theorem 4.5 The Definite Integral as the Area of a Region

3. Fundamental Theorem of Calculus
4.3

4. The Fundamental Theorem of Calculus
If a function f is continuous on the closed [a,b] and F is an antiderivative of f on the interval [a,b], then:

5. FTC and the Constant of Integration
It is not necessary to include a constant of integration C in the antiderivative because:

6. FTC Example

7. Using FTC to Find Area
Example 3 in the text: 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
Note that all of the function is above the x-axis in the interval
Start by integrating the function over the closed interval [0,2]
Then find the antiderivative and apply the FTC
Simplify.

8. Theorem 4.4 Continuity Implies Integrability

9. Theorem 4.6 Additive Interval Property

10. Absolute Value Integration Example:
Example 2 in the text: keep in mind the definition of absolute value and where your function is positive and where it would be negative:
From this, you can rewrite the integral in two parts:

11. Theorem 4.7 Properties of Definite Integrals

12. Definitions of Two Special Definite Integrals

13. Average Value of a Function
Julia S. Lucas

14. Definition of the Average Value of a Function on an Interval and Figure 4.32

15. Let’s get started:
You already know about functions and how to take the average of some finite set.
Today we’re going to take the average over infinitely many values (those that the function takes on over some interval)…which means CALCULUS!
Before I show how to do this…let’s talk about WHY we might want to do this?

16. Consider the following picture:
How high would the water level be if the waves all settled?

17. Okay! So, now that you have seen that this an interesting question...
Let’s forget about real life, and...
Do Some Math

18. Suppose we have a “nice”function and we need to find its average value over the interval [a,b].

19. Let’s apply our knowledge of how to find the
average over a finite set of values to this problem:
First, we partition the interval [a,b] into n subintervals of
equal length to get back to the finite situation:
In the above graph, we have n=8

20. Let us set up our notation:
comes from the i-th interval
Now we can get an estimate for the
average value:

21. Let’s try to clean this up a little:
Since

22. Take Limits!!! In a more condensed form, we now get:
But we want to get out of the finite,
and into the infinite!
How do we do this?
Take Limits!!!

23. In this way, we get the average value of f(x) over the interval [a,b]:
So, if f is a “nice” function (i.e. we
can compute its integral) then we have
a precise solution to our problem.

24. Let’s look back at our graph:

25. So we’ve solved our problem!
If I give you the equation
and ask you to find it’s average value
over the interval [0,2], you’ll all say
NO PROBLEM!!!!
the answer is…

27. Now we can answer our fish tank question
Now we can answer our fish tank question! (That is, if the waves were described by an integrable function)

28. Theorem 4.10 Mean Value Theorem for Integrals and Figure 4.30

29. Finding the x values where we GET the
average value of the function.
First take the equation
and find it’s average value
over the interval [0,2],
NO PROBLEM!!!!
the answer is…

31. Finding the x values where we GET the
average value of the function.
NOW take the equation
and it’s average value
over the interval [0,2], average f = 5
and set that equal to the function…
And then solve for x…
NO PROBLEM!!!!
the answer is…

32. Mean Value Theorem (find the value of x that gives you the average value of your function)

33. Average Value of a Function
Definition:
If f is integrable on the closed interval [a,b], then the average value of f on the interval is:

34. Average Value Example
Find the average value of

35. Mean Value Theorem
If f is continuous on the closed interval [a,b], then there exists a number c in the closed interval [a,b] such that:

36. Mean Value Theorem Example:
Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval:
Now we need to find the x coordinate where we get this average y value:

37. MVT-I
The mean value theorem does not specify how to find c, just that there exists at least one number c in the interval that will give you the average value of the function.

38. Theorem 4.8 Preservation of Inequality

39. The Second Fundamental Theorem of Calculus
4.4

40. Second Fundamental Theorem of Calculus
Earlier you saw that the definite integral of f on the interval [a,b] was defined using the constant b as the upper limit of integration and x as the variable of integration. However, a slightly different situation may arise in which the variable x is used as the upper limit of integration. To avoid the confusion of using x in two different ways, t is temporarily used as the variable of integration

41. Theorem 4.11 The Second Fundamental Theorem of Calculus

42. Definite Integral diagrams

43. The Definite Integral as a Function
You can think of the function F(x) as accumulating the area under the curve f(t)=cost from t=0 to t=x. For x=0, the area is 0 and F(x)=0. for x=pi/2, F(pi/2)=1 gives the accumulated area under the cosine curve on the entire interval [0,pi/2]. This interpretation of an integral as an accumulation function is used often in applications of integration. See p. 288 for a graphical example of this.

44. Figure 4.35

45. The Second Fundamental Theorem of Calculus
If f is continuous on an open interval I containing a, then, for every x in the interval,
The proof of this is on p. 289 in the text.

46. Using the SFTC
Note that is continuous on the entire real line. So, using the SFTC you can write
The differentiation shown in tthis example is a straightforward application of the SFTC. The next example shows an application of this combined with the chain rule to find the derivative of a function

47. Using the SFTC Chain Rule Definition of dF/du
Substitute the integral for F(x)
Substitute u for x3
Apply the SFTC
Rewrite as a function of x.

48. Find 𝐹′(𝑥) if 𝐹 𝑥 = 0 𝑥 2 𝑠𝑖𝑛 𝜃 2 𝑑𝜃
What is u and du? 𝑢= 𝑥 2 , 𝑑𝑢=2𝑥 𝑑 𝑑𝑢 0 𝑢 𝑠𝑖𝑛 𝜃 2 𝑑𝜃 𝑑𝑢 𝑑𝑥 =𝑠𝑖𝑛 𝑢 2 𝑑𝑢 =sin⁡( 𝑥 2 ) 2 (2𝑥) =𝑠𝑖𝑛 𝑥 4 (2𝑥)

49. U-Substitution

50. Antidifferentiation of a Composite Function
Let g be a function whose range is an interval I and let f be a function that is continuous on I. If g is differentiable on its domain and F is an antiderivative of f on I, then
If u = g(x), then du = g’(x)dx and

51. Pattern Recognition
In this section you will study techniques for integrating composite functions.
This is split into two parts, pattern recognition and change of variables.
u-substitution is similar to the techniques used for the chain rule in differentiation.

52. Recognizing Patterns

53. Pattern Recognition for Finding the Antiderivative
Let g(x)=5x and we have g’(x)=5
So we have f(g(x))=f(5x)=cos5x
From this, you can recognize that the integrand follows the f(g(x))g’(x) pattern. Using the trig integration rule, we get
You can check this by differentiating the answer to obtain the original integrand.

54. Multiplying and Dividing by a Constant
Example 3 p. 297

55. Change of Variables Example 4 p. 298

56. Change of Variables
Example 5 p. 298

57. Change of Variables Example
Example 6 p. 299

58. Guidelines for Making a Change of Variables
Choose a substitution u=g(x). Usually it is best to choose the INNER part of a composite function, such as a quantity raised to a power.
Compute du=g’(x)dx
Rewrite the integral in terms of the variable u
Find the resulting integral in terms of u
Replace u by g(x) to obtain an antiderivative in terms of x.
Check your answer by differentiating.

59. General Power Rule for Integration
Theorem: The General Power Rule for Integration
If g is a differentiable function of x, then
Equivalently, if u=g(x), then

60. Theorem 4.14 Change of Variables for Definite Integrals

61. Change of Variables for Definite Integrals
Theorem 4.14 p 301

62. Definite Integrals and Change of Variables
Example 8 p. 301

63. Definite Integrals and Change of Variables
Example 9 p. 302

64. Theorem 4.15 Integraion of Even and Odd Functions and Figure 4.39

65. Integration of Even and Odd Functions
P. 303

66. Integration of an Odd Function
Example 10 p. 303

67. Figure 4.41

68. Theorem 4.16 The Trapezoidal Rule

69. Theorem 4.17 Integral of p(x) =Ax2 + Bx + C

70. Theorem 4.18 Simpson's Rule (n is even)

71. Theorem 4.19 Errors in the Trapezoidal Rule and Simpson's Rule