1. Antidifferentiation and Integration Suggested Time:19 Hours
Techniques of antidifferentiation
Initial Value Problems
Connection to area under a curve
Fundamental Theorem of Calculus
Area Between 2 curves

2. Techniques of Antidifferentiation
Antidifferentation is the opposite of differentiation.
It is the process to “undo” the process of differentiation

3. Consider a differentiable function F(x) where
What are some possibilities for F(x)?
F(x) = x2
F(x) = _______
The General Solution is F(x) = x2 + c, where c is some constant.
These are all
Particular Solutions

4. Using graphing technology, graph
F(x) = x2 + 8 and G(x) = x2 + 2
on the same axis and construct the tangent lines at x = 1.
What is the equation of the tangent line to the curve F(x) at x = 1?
What is the equation of the tangent line to the curve G(x) at x = 1?

6. What is the simplified derivative?
C) Why is ?
They both simplify to the same derivative.
What is the simplified derivative?
D) Why do functions, which vary by a constant, have the same slope (i.e., m = 2) for tangents drawn at the same value of x?
Because they have the same derivative function.
E) Why are the graphs of antiderivatives of a given function vertical translations of each other (i.e., family of curves)?
Because the only vary by a constant.

7. So, lets consider F(x) is the antiderivative of f(x).
For each derivative function f(x), there can be more than one antiderivative function, F(x).
But they all differ by a constant.
The process of finding F(x), given f(x), is called antidifferentation (undoing the derivative) or integration

8. We represent this process by:
which is equivalent to
f(x) is called the integrand
dx indicates the variable of integration.
So when you are asked, for example, to find all the functions whose derivative is 2x, you can now express this as:
Integral sign

9. Note: 1. 2.

10. Example: Find 1. 2. 3.

11. Davin stated that the solution for
is the same as
Do you agree or disagree with this statement. Justify your answer.

12. What are some possible antiderivatives for the following functions
What are some possible antiderivatives for the following functions? Verify your results using differentiation.
(i) f (x) = 5 (ii) g(x) = 4x (iii) h(x) = 3x2 (iv) k(x) = 3x2 + 4x + 5

13. Most of our differentiation rules can be rewritten into integration rules.
1. Constant Rule 2.
A constant c
But c does not have to equal k
+ c

14. 3. Constant Rule II
Integration Rule:

15. Example:
Find:

16. 4. Power Rule
+ c
Examples:

17. 2. 3.

18. 5. Addition Subtraction Rule
Example
Integrate:

19. Try this one
We need to wait for the next Unit (Exponential and Logarithmic Functions) before we can do this antiderivative!

20. NOTE:
There are no Product or Quotient Rules for antidifferentiation.

21. Transcendental Rules (NOT IN 3208, but good to know for post secondary Calculus)

23. Examples. Find the general antiderivative for the following:1. 2.

24. 3. 4. 5.

25. Examples. Integrate! 1. 2.

26. 3. 4. 5.

27. 6. 7.

28. Text Page 194
# 1-7

29. Initial Condition Problems
If we are given initial conditions we can find particular anti-derivatives rather than general ones.
We can find a value for c.

30. Example: 1. Find f(x) if and f(2) = 9 Because of this part we are
looking for a particular derivative
1. Find f(x) if and f(2) = 9

31. 2. Find h(t) if and h(0) =2

32. 3. Find y if and (4, -1) is a point on the graph of y.

33. 4. Find f(x) if and the slope of the tangent line at the point (2, 3) is -8
Note: There are two conditions given in this problem

34. Practice:
A) Find f(x) if and f(0) = 0
and
B) If , find the equation for g(x)
C) Find f(x) if and f(1) = f(-1) = 5  

35. There are many “initial value” application problems
Biologists may want to determine the future population given the current rate of population growth.
A banker may want to determine future value of a mortgage given current interest rates.
For the purpose of this course, the problems will focus exclusively on motion problems.

36. Velocity and Acceleration Problems
Recall: where s(t) is a position function.
And
Thus and

37. Example 1:
A particle moves along a straight line with an acceleration of a (t) = t2 + 2.
If at t = 0 the particle is stationary, write an equation describing its velocity, v (t).

38. Example 2:
The velocity of a ball being thrown in the air from an initial height of 1 m is given by
v(t) = -9.8t + 12, where t is the time in seconds.
A) Determine the quadratic function that models the height of the ball after t seconds.

39. Example 2:
B)Use this function found in A) to calculate the height of the ball after 2.5 seconds.

40. Example 3:
A rock is dropped from a cliff. If it takes 4.0s to hit the bottom of the cliff:
A) How high is the cliff?
B) What was the rock’s velocity when it hit?
Solution: For falling objects the earth gravity causes an acceleration of
In the USA

41. Since the rock was dropped v(0)= 0m/s
And

43. Practice:
A) A particle is accelerated in a line so that its acceleration, in
Determine the position of the particle at time, t, if s (0) = v (0) = 0.

44. B) A ball is thrown straight up from a height of 20 m with an initial velocity of 40 m/s and acceleration is -10 m/s2 (i) At what time will the ball reach its maximum height? (ii) At what time will it hit the ground? (iii) What is the velocity of the ball when it strikes the ground? (iv) What is the total distance travelled by the ball?

45. B) A ball is thrown straight up from a height of 20 m with an initial velocity of 40 m/s and acceleration is -10 m/s2
(i) At what time will the ball reach its maximum height?
Find the velocity function

46. B) A ball is thrown straight up from a height of 20 m with an initial velocity of 40 m/s and acceleration is -10 m/s2
(ii) At what time will it hit the ground?
Find the displacement function

47. (iii) What is the velocity of the ball when it strikes the ground
(iii) What is the velocity of the ball when it strikes the ground? (iv) What is the total distance travelled by the ball?

48. Area under a curve

49. AREA and the Definite Integral
Estimate an area using a finite sum.
Determine the area using the infinite Riemann sum.
Convert a Riemann sum to a definite integral.
Using definite integrals, determine the area under a polynomial function from x = a to x = b.
Calculate the definite integral of a function over an interval [a, b]
Determine the area between two polynomial functions

50. Area
Recall from Physics in 1206 Science that you can calculate the displacement of an object from its velocity vs time graph. How is this done?
Find the area under the curve!

51. Example: Find the displacement from t=0 to t=5
Find the area of the trapezoid

52. However: What is v(t) in this case? Find s(t), provided s(0) = 0.
Find the equation of the line!
Find s(t), provided s(0) = 0.
Find s(5)

53. SO, we can see that the integral of a function is related to the area under the function.
In this section we are going to develop ways of finding areas under functions (not all areas will be as easy as the last one).
And then, ultimately show how this can be calculated using a “definite” integral.

54. Examples
1. Determine the area, for example, of the shaded region below the line y = 2x + 3 and above the x-axis from x = 0 to x = 3.
Students may notice that the total area can be viewed as the area of a
trapezoid or as a combination of the areas of a rectangle and a triangle.

55. Note: The total area could have been viewed as the area of :
a trapezoid
or as a combination of the areas of a rectangle and a triangle.
This would be the case for any linear function.
The challenge arises when the function is something other than linear.

56. 2. Determine the area under the curve
y = x2 and above the x-axis from x=0 to x=3.
We can estimate the area by finding the area of a shape that resembles the desired area
This would be a triangle in this case.

57. Is this a good approximation?
Is it too big or too small?
It is too big!
The triangle includes area above the parabola.
How can we get a better approximation?
We can divide the area into many smaller areas in the shape of rectangles (or trapezoids).
The area of each rectangle is calculated and summed.
This strategy only produces an approximation to the area.
This approximation, however, improves as the number of rectangles increase.

58. Lets estimate the area with 3 rectangles
We will first subdivide the area using partitions (or subintervals) taken along the x-axis
How wide will each rectangle be if they are all equal width?
the width of each subinterval, is denoted
We draw rectangles using the function value of the right endpoint as the height of the rectangle.

59. Note:
We could use the left endpoint or the midpoint of each interval to determine the height of the rectangles.
In fact we could use any point in the interval to determine the intervals height.
In 3208, we will only use the right endpoint of each interval

60. So the area is approximated by adding the area of all of these rectangles together.
Remember the area for a rectangle??
A =width x height
Therefore the area is:

61. 3. Determine the area under the curve
y = x2 and above the x-axis from x=0 to x=3.
This time lets use more rectangles.
SIX

62. What is the total width of the interval?
If there are six rectangles, what is the width of each rectangle?
Which values should be used to determine the height of each rectangle?

63. What is the area of each rectangle?
What is the sum of all six rectangles?

64. Number of sub-intervals (n) Sum of the areas of the rectangles
With the aid of technology, generate the area approximations using sub-intervals of 12, 24, 48, and 80.
Number of sub-intervals (n)
Sum of the areas of the rectangles 3 6 12 24 48 80

65. There must be an APP for this!

66. The chart illustrates that, as the number of sub-intervals increase, there is an increase in the accuracy of the estimation.
As n increases, the sums converge to a limit, which is the whole idea of integral calculus.
What is the limit??
Remember this is an educated guess at the area.
Before we can calculate this limit, we must be introduced to sigma notation.

68. In General
Suppose we wanted to find the area under this function (curve) y = f(x) from a to b. a b

69. We will first subdivide the area using partitions (or subdivisions) taken along the x-axis
We draw rectangles using the function value of the right endpoint as the height of the rectangle. a b

70. We will first subdivide the area using partitions (or subdivisions) taken along the x-axis
We draw rectangles using the function value of the right endpoint as the height of the rectangle. a b

71. So the area is approximated by adding the area of all of these rectangles together.
Remember the area for a rectangle??
A =width x height
Therefore the area becomes

72. Note: If all the n rectangles were all of equal width, each width would be:
The area
becomes

73. Example: Calculate the area under y = x2 from 1 to 3 using
A) 4 equal subdivisions

74. Example: Calculate the area under y = x2 from 1 to 3 using
B) 8 equal subdivisions, and the right endpoint for height.

75. Practice: Calculate the area under y = x2 + 1 from 1 to 3 using
10 equal subdivisions.

76. Practice: Calculate the area under y = sin x from 0 to p using
6 equal subdivisions.

77. How can we get better approximations?
More intervals

78. A lot more intervals!
In general, as the number of rectangles increase, the estimate gets closer to the actual value.
In fact, if n is the number of rectangles, then as the width of each rectangle approaches ____
Also, the limit of the sum of the rectangles as produces the actual area.

79. Find the limit of the sum of the area of ALL of the n strips
In general, to find the area under y = f(x) from a to b where f(x) is a continuous function and f(x) > 0 (positive) we:
Subdivide the area (region) into n strips of equal width Dx.
Find the limit of the sum of the area of ALL of the n strips

80. Ai Represents the area of the ith rectangle. Ai = l x w Ai = f(xi) Dx
w = Dx
l = f(xi)
Ai = f(xi) Dx

81. Recall

82. xi
Represents the x value used to find the height (length) of the ith rectangle.
xi can be the:
Right endpoint where

83. Example: Calculate the area under y = x2 from 0 to 2 using
An infinite number of subdivisions, and the right endpoint for height.

86. Sigma Notation and Series
Definition: A series is the SUM of a sequence of numbers.
1, 4, 9, 16 is a sequence.
is a series
What is the formula for the sequence?
What is the formula for the series?

87. This format is cumbersome, so a new type of notation is used to simplify expressing series.
This notation is called sigma (S) notation.
Sigma (S) is the Greek capital letter S for series (Sum).
The last series can be expressed as:

88. Summation rules
1. Constant Rule: A) If c is a constant then B)

89. 2. 3. 4. 5.

90. Example: Evaluate

92. Area examples:
1. Find the area under y = x3 from x = 1 to x = 4

94. 2. Find the area under y = (x + 1)2 from x = 0 to x = 2

95. 2. Find the area under y = (x + 1)2 from x = 0 to x = 2

98. Evaluate the Riemann sum for the function, f (x) = x3 - 4x, on the interval [0, 2] with n equally spaced subintervals.

99. This value represents the signed area, where the region located:
The Riemann sum is -4.
The area is actually 4, and the negative sign is an indication that the area lies below the x-axis.
This value represents the signed area, where the region located:
below the x-axis will be counted as negative,
and the region above will be counted as positive.

100. Example: Identify which areas are positive and which are negative.

101. The Riemann sum is the sum of the areas of the rectangles that lie above the x-axis and the negative of the areas that lie below the x-axis.
This difference between the positive and negative contributions is called the net signed area.

102. Note:
The net signed area can be positive, negative, or zero.
If the net signed area is negative then there is more area below the x-axis than above.
If the net signed area is positive then there is more area above the x-axis than below.
If the net signed area is zero then the area above the x-axis is equal to the area below the x-axis.

103. Use the Riemann sum to estimate the area under the curve y = 3x - x2 from 0 ≤ x ≤ 5.

104. Why is the value of the Riemann sum negative?

105. Definite Integral

106. Definition:
Let y = f(x) be a continuous function defined on the closed interval [a, b].
The definite integral of f(x) from a to b is:
where
xi is some x value (any value) in the ith interval
f(x) is the integrand
a and b are limits of integration
a is the lower limit and b is the upper limit
dx gives the variable which is being integrated.
Riemann
Sum

107. What is the difference between and ?
The indefinite integral is a function.
The definite integral is number.

108. Uses:
Area.
If f(x) is positive on [a,b] then represents the area under the curve f(x) between a and b.
In general, gives the area between f(x) and the x-axis from a to b.
If the area is below the x-axis, the area is said to be negative.

109. Other uses Finding regions of area between 2 functions.
Finding the volumes of solids of
Revolution
Known cross sections
Finding the arclength of a function
The distance travelled by a particle along a line.
The average value of a function

110. Other uses Surface area of solid figures. Centre of mass of objects.
Work done on an object

111. All of the RED stuff will be done in the next Calculus course.
First though we need to be able to evaluate these definite integrals.

112. Riemann Sum way.
Find:

113. Riemann Sum way.
Find:

114. Quick way. (2nd Fundamental Theorem of Calculus)
If f(x) is a continuous function on the interval [a, b] then
where F(x) is any anti-derivative of f(x).
Notation:

115. Quick way. (2nd Fundamental Theorem of Calculus)
Find:
Same answer as with Riemann sums, but slightly easier!
The constant of integration (c) can be dropped because they will cancel out
Or, since F(x) is any antiderivative, take c = 0

116. Examples:
1. Find

117. Examples:
2.A)
This represents a signed area of 4 units2 above the x-axis.
f(x) > 0

118. Examples:
2.B)
This represents a signed area of 4 units2 below the x-axis.
f(x) < 0

119. Examples:
2.C)
This represents a signed area of 0.

120. This would be a good opportunity to compare net area and total area.
If the problem asks for the net area (or signed area) the answer can be obtain by evaluating the definite integral.
If asked to find the total area, the area must be broken into 2 parts:
The area above the x-axis
The area below the x-axis
And then combine the 2 areas.
In the last problem the total area would be ______

121. Examples: 3. Consider f(x) = 1 - 2x
A) Determine signed area between [0, 2]

122. Examples: 3. Consider f(x) = 1 - 2x
B) Determine total area between [0, 2]
You need to determine where the function equals zero

124. Note:
If the functions represents a velocity function then the signed area is ____________
And the total area is _____________

125. Consider the velocity function
A) Find the displacement from t = 0 to t = 2

126. Consider the velocity function
B) Find the total distance travelled from t = 0 to t = 2

127. Find the area between and the x-axis from -3 to 3.

128. Find the area between and the x-axis from -3 to 3.

129. Evaluate A) Using the Fundamental Theorem
B) Riemann sums, with right endpoints
C) Interpret the integral as an area problem, and illustrate with a diagram
TAKE HOME QUIZ!!!!

133. REMOVE THE ABSOLUTE VALE SIGN
We need to break the definite
integral into 2 different integrals
based on the location of the
sharp point.

134. Area between Curves (Area of Regions)
y = g(x)
Suppose we wanted to find the area between f and g over the interval from a to b.
y = f(x)

135. Take rectangles:
Since
For all values in
[a, b]

137. Example: 1. Find the region below
Find the functions first!
f(x) = 4 – x2
g(x) = 2x + 1
Where do the functions intersect?

138. 2. Find the area bounded by y = x2 + 2, y = -x, x = 0, and x = 2

139. 3. Find the area between y = 2 - x2, y = x.

140. Sometimes when finding the area between functions, the graphs criss-cross each other.
We find the bounded area by breaking the integral into parts

141. 4. Determine the total area between the functions f (x) = x3 - 5x2 - x + 15 and g(x) = x2 - 4x + 5.

142. 4. Determine the total area between the functions f (x) = x3 - 5x2 - x + 15 and g(x) = x2 - 4x + 5.
Find the x-values of the points of intersection of the two functions by setting f (x) = g(x) (i.e., limits of integration).

143. Use the graph if you have one
To determine which function is above the other in each interval you can:
Use the graph if you have one
Or substitute a x value from each interval into both of the functions.
The larger function value is the upper function.

144. Practice
1. Determine the area bounded by: y = x2 + 2, y = -x, x = 0 and x = 2. 2.Determine the area of the region between the given curves: y = 2 + x2 and y = 2x2 - 7