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INSTRUCTORS Raffi Hovasapian John Zhu
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Raffi Hovasapian

Raffi Hovasapian

Example Problems for Area Under a Curve

Slide Duration:

Table of Contents

I. Limits and Derivatives
Overview & Slopes of Curves

42m 8s

Intro
0:00
Overview & Slopes of Curves
0:21
Differential and Integral
0:22
Fundamental Theorem of Calculus
6:36
Differentiation or Taking the Derivative
14:24
What Does the Derivative Mean and How do We Find it?
15:18
Example: f'(x)
19:24
Example: f(x) = sin (x)
29:16
General Procedure for Finding the Derivative of f(x)
37:33
More on Slopes of Curves

50m 53s

Intro
0:00
Slope of the Secant Line along a Curve
0:12
Slope of the Tangent Line to f(x) at a Particlar Point
0:13
Slope of the Secant Line along a Curve
2:59
Instantaneous Slope
6:51
Instantaneous Slope
6:52
Example: Distance, Time, Velocity
13:32
Instantaneous Slope and Average Slope
25:42
Slope & Rate of Change
29:55
Slope & Rate of Change
29:56
Example: Slope = 2
33:16
Example: Slope = 4/3
34:32
Example: Slope = 4 (m/s)
39:12
Example: Density = Mass / Volume
40:33
Average Slope, Average Rate of Change, Instantaneous Slope, and Instantaneous Rate of Change
47:46
Example Problems for Slopes of Curves

59m 12s

Intro
0:00
Example I: Water Tank
0:13
Part A: Which is the Independent Variable and Which is the Dependent?
2:00
Part B: Average Slope
3:18
Part C: Express These Slopes as Rates-of-Change
9:28
Part D: Instantaneous Slope
14:54
Example II: y = √(x-3)
28:26
Part A: Calculate the Slope of the Secant Line
30:39
Part B: Instantaneous Slope
41:26
Part C: Equation for the Tangent Line
43:59
Example III: Object in the Air
49:37
Part A: Average Velocity
50:37
Part B: Instantaneous Velocity
55:30
Desmos Tutorial

18m 43s

Intro
0:00
Desmos Tutorial
1:42
Desmos Tutorial
1:43
Things You Must Learn To Do on Your Particular Calculator
2:39
Things You Must Learn To Do on Your Particular Calculator
2:40
Example I: y=sin x
4:54
Example II: y=x³ and y = d/(dx) (x³)
9:22
Example III: y = x² {-5 <= x <= 0} and y = cos x {0 < x < 6}
13:15
The Limit of a Function

51m 53s

Intro
0:00
The Limit of a Function
0:14
The Limit of a Function
0:15
Graph: Limit of a Function
12:24
Table of Values
16:02
lim x→a f(x) Does not Say What Happens When x = a
20:05
Example I: f(x) = x²
24:34
Example II: f(x) = 7
27:05
Example III: f(x) = 4.5
30:33
Example IV: f(x) = 1/x
34:03
Example V: f(x) = 1/x²
36:43
The Limit of a Function, Cont.
38:16
Infinity and Negative Infinity
38:17
Does Not Exist
42:45
Summary
46:48
Example Problems for the Limit of a Function

24m 43s

Intro
0:00
Example I: Explain in Words What the Following Symbols Mean
0:10
Example II: Find the Following Limit
5:21
Example III: Use the Graph to Find the Following Limits
7:35
Example IV: Use the Graph to Find the Following Limits
11:48
Example V: Sketch the Graph of a Function that Satisfies the Following Properties
15:25
Example VI: Find the Following Limit
18:44
Example VII: Find the Following Limit
20:06
Calculating Limits Mathematically

53m 48s

Intro
0:00
Plug-in Procedure
0:09
Plug-in Procedure
0:10
Limit Laws
9:14
Limit Law 1
10:05
Limit Law 2
10:54
Limit Law 3
11:28
Limit Law 4
11:54
Limit Law 5
12:24
Limit Law 6
13:14
Limit Law 7
14:38
Plug-in Procedure, Cont.
16:35
Plug-in Procedure, Cont.
16:36
Example I: Calculating Limits Mathematically
20:50
Example II: Calculating Limits Mathematically
27:37
Example III: Calculating Limits Mathematically
31:42
Example IV: Calculating Limits Mathematically
35:36
Example V: Calculating Limits Mathematically
40:58
Limits Theorem
44:45
Limits Theorem 1
44:46
Limits Theorem 2: Squeeze Theorem
46:34
Example VI: Calculating Limits Mathematically
49:26
Example Problems for Calculating Limits Mathematically

21m 22s

Intro
0:00
Example I: Evaluate the Following Limit by Showing Each Application of a Limit Law
0:16
Example II: Evaluate the Following Limit
1:51
Example III: Evaluate the Following Limit
3:36
Example IV: Evaluate the Following Limit
8:56
Example V: Evaluate the Following Limit
11:19
Example VI: Calculating Limits Mathematically
13:19
Example VII: Calculating Limits Mathematically
14:59
Calculating Limits as x Goes to Infinity

50m 1s

Intro
0:00
Limit as x Goes to Infinity
0:14
Limit as x Goes to Infinity
0:15
Let's Look at f(x) = 1 / (x-3)
1:04
Summary
9:34
Example I: Calculating Limits as x Goes to Infinity
12:16
Example II: Calculating Limits as x Goes to Infinity
21:22
Example III: Calculating Limits as x Goes to Infinity
24:10
Example IV: Calculating Limits as x Goes to Infinity
36:00
Example Problems for Limits at Infinity

36m 31s

Intro
0:00
Example I: Calculating Limits as x Goes to Infinity
0:14
Example II: Calculating Limits as x Goes to Infinity
3:27
Example III: Calculating Limits as x Goes to Infinity
8:11
Example IV: Calculating Limits as x Goes to Infinity
14:20
Example V: Calculating Limits as x Goes to Infinity
20:07
Example VI: Calculating Limits as x Goes to Infinity
23:36
Continuity

53m

Intro
0:00
Definition of Continuity
0:08
Definition of Continuity
0:09
Example: Not Continuous
3:52
Example: Continuous
4:58
Example: Not Continuous
5:52
Procedure for Finding Continuity
9:45
Law of Continuity
13:44
Law of Continuity
13:45
Example I: Determining Continuity on a Graph
15:55
Example II: Show Continuity & Determine the Interval Over Which the Function is Continuous
17:57
Example III: Is the Following Function Continuous at the Given Point?
22:42
Theorem for Composite Functions
25:28
Theorem for Composite Functions
25:29
Example IV: Is cos(x³ + ln x) Continuous at x=π/2?
27:00
Example V: What Value of A Will make the Following Function Continuous at Every Point of Its Domain?
34:04
Types of Discontinuity
39:18
Removable Discontinuity
39:33
Jump Discontinuity
40:06
Infinite Discontinuity
40:32
Intermediate Value Theorem
40:58
Intermediate Value Theorem: Hypothesis & Conclusion
40:59
Intermediate Value Theorem: Graphically
43:40
Example VI: Prove That the Following Function Has at Least One Real Root in the Interval [4,6]
47:46
Derivative I

40m 2s

Intro
0:00
Derivative
0:09
Derivative
0:10
Example I: Find the Derivative of f(x)=x³
2:20
Notations for the Derivative
7:32
Notations for the Derivative
7:33
Derivative & Rate of Change
11:14
Recall the Rate of Change
11:15
Instantaneous Rate of Change
17:04
Graphing f(x) and f'(x)
19:10
Example II: Find the Derivative of x⁴ - x²
24:00
Example III: Find the Derivative of f(x)=√x
30:51
Derivatives II

53m 45s

Intro
0:00
Example I: Find the Derivative of (2+x)/(3-x)
0:18
Derivatives II
9:02
f(x) is Differentiable if f'(x) Exists
9:03
Recall: For a Limit to Exist, Both Left Hand and Right Hand Limits Must Equal to Each Other
17:19
Geometrically: Differentiability Means the Graph is Smooth
18:44
Example II: Show Analytically that f(x) = |x| is Nor Differentiable at x=0
20:53
Example II: For x > 0
23:53
Example II: For x < 0
25:36
Example II: What is f(0) and What is the lim |x| as x→0?
30:46
Differentiability & Continuity
34:22
Differentiability & Continuity
34:23
How Can a Function Not be Differentiable at a Point?
39:38
How Can a Function Not be Differentiable at a Point?
39:39
Higher Derivatives
41:58
Higher Derivatives
41:59
Derivative Operator
45:12
Example III: Find (dy)/(dx) & (d²y)/(dx²) for y = x³
49:29
More Example Problems for The Derivative

31m 38s

Intro
0:00
Example I: Sketch f'(x)
0:10
Example II: Sketch f'(x)
2:14
Example III: Find the Derivative of the Following Function sing the Definition
3:49
Example IV: Determine f, f', and f'' on a Graph
12:43
Example V: Find an Equation for the Tangent Line to the Graph of the Following Function at the Given x-value
13:40
Example VI: Distance vs. Time
20:15
Example VII: Displacement, Velocity, and Acceleration
23:56
Example VIII: Graph the Displacement Function
28:20
II. Differentiation
Differentiation of Polynomials & Exponential Functions

47m 35s

Intro
0:00
Differentiation of Polynomials & Exponential Functions
0:15
Derivative of a Function
0:16
Derivative of a Constant
2:35
Power Rule
3:08
If C is a Constant
4:19
Sum Rule
5:22
Exponential Functions
6:26
Example I: Differentiate
7:45
Example II: Differentiate
12:38
Example III: Differentiate
15:13
Example IV: Differentiate
16:20
Example V: Differentiate
19:19
Example VI: Find the Equation of the Tangent Line to a Function at a Given Point
12:18
Example VII: Find the First & Second Derivatives
25:59
Example VIII
27:47
Part A: Find the Velocity & Acceleration Functions as Functions of t
27:48
Part B: Find the Acceleration after 3 Seconds
30:12
Part C: Find the Acceleration when the Velocity is 0
30:53
Part D: Graph the Position, Velocity, & Acceleration Graphs
32:50
Example IX: Find a Cubic Function Whose Graph has Horizontal Tangents
34:53
Example X: Find a Point on a Graph
42:31
The Product, Power & Quotient Rules

47m 25s

Intro
0:00
The Product, Power and Quotient Rules
0:19
Differentiate Functions
0:20
Product Rule
5:30
Quotient Rule
9:15
Power Rule
10:00
Example I: Product Rule
13:48
Example II: Quotient Rule
16:13
Example III: Power Rule
18:28
Example IV: Find dy/dx
19:57
Example V: Find dy/dx
24:53
Example VI: Find dy/dx
28:38
Example VII: Find an Equation for the Tangent to the Curve
34:54
Example VIII: Find d²y/dx²
38:08
Derivatives of the Trigonometric Functions

41m 8s

Intro
0:00
Derivatives of the Trigonometric Functions
0:09
Let's Find the Derivative of f(x) = sin x
0:10
Important Limits to Know
4:59
d/dx (sin x)
6:06
d/dx (cos x)
6:38
d/dx (tan x)
6:50
d/dx (csc x)
7:02
d/dx (sec x)
7:15
d/dx (cot x)
7:27
Example I: Differentiate f(x) = x² - 4 cos x
7:56
Example II: Differentiate f(x) = x⁵ tan x
9:04
Example III: Differentiate f(x) = (cos x) / (3 + sin x)
10:56
Example IV: Differentiate f(x) = e^x / (tan x - sec x)
14:06
Example V: Differentiate f(x) = (csc x - 4) / (cot x)
15:37
Example VI: Find an Equation of the Tangent Line
21:48
Example VII: For What Values of x Does the Graph of the Function x + 3 cos x Have a Horizontal Tangent?
25:17
Example VIII: Ladder Problem
28:23
Example IX: Evaluate
33:22
Example X: Evaluate
36:38
The Chain Rule

24m 56s

Intro
0:00
The Chain Rule
0:13
Recall the Composite Functions
0:14
Derivatives of Composite Functions
1:34
Example I: Identify f(x) and g(x) and Differentiate
6:41
Example II: Identify f(x) and g(x) and Differentiate
9:47
Example III: Differentiate
11:03
Example IV: Differentiate f(x) = -5 / (x² + 3)³
12:15
Example V: Differentiate f(x) = cos(x² + c²)
14:35
Example VI: Differentiate f(x) = cos⁴x +c²
15:41
Example VII: Differentiate
17:03
Example VIII: Differentiate f(x) = sin(tan x²)
19:01
Example IX: Differentiate f(x) = sin(tan² x)
21:02
More Chain Rule Example Problems

25m 32s

Intro
0:00
Example I: Differentiate f(x) = sin(cos(tanx))
0:38
Example II: Find an Equation for the Line Tangent to the Given Curve at the Given Point
2:25
Example III: F(x) = f(g(x)), Find F' (6)
4:22
Example IV: Differentiate & Graph both the Function & the Derivative in the Same Window
5:35
Example V: Differentiate f(x) = ( (x-8)/(x+3) )⁴
10:18
Example VI: Differentiate f(x) = sec²(12x)
12:28
Example VII: Differentiate
14:41
Example VIII: Differentiate
19:25
Example IX: Find an Expression for the Rate of Change of the Volume of the Balloon with Respect to Time
21:13
Implicit Differentiation

52m 31s

Intro
0:00
Implicit Differentiation
0:09
Implicit Differentiation
0:10
Example I: Find (dy)/(dx) by both Implicit Differentiation and Solving Explicitly for y
12:15
Example II: Find (dy)/(dx) of x³ + x²y + 7y² = 14
19:18
Example III: Find (dy)/(dx) of x³y² + y³x² = 4x
21:43
Example IV: Find (dy)/(dx) of the Following Equation
24:13
Example V: Find (dy)/(dx) of 6sin x cos y = 1
29:00
Example VI: Find (dy)/(dx) of x² cos² y + y sin x = 2sin x cos y
31:02
Example VII: Find (dy)/(dx) of √(xy) = 7 + y²e^x
37:36
Example VIII: Find (dy)/(dx) of 4(x²+y²)² = 35(x²-y²)
41:03
Example IX: Find (d²y)/(dx²) of x² + y² = 25
44:05
Example X: Find (d²y)/(dx²) of sin x + cos y = sin(2x)
47:48
III. Applications of the Derivative
Linear Approximations & Differentials

47m 34s

Intro
0:00
Linear Approximations & Differentials
0:09
Linear Approximations & Differentials
0:10
Example I: Linear Approximations & Differentials
11:27
Example II: Linear Approximations & Differentials
20:19
Differentials
30:32
Differentials
30:33
Example III: Linear Approximations & Differentials
34:09
Example IV: Linear Approximations & Differentials
35:57
Example V: Relative Error
38:46
Related Rates

45m 33s

Intro
0:00
Related Rates
0:08
Strategy for Solving Related Rates Problems #1
0:09
Strategy for Solving Related Rates Problems #2
1:46
Strategy for Solving Related Rates Problems #3
2:06
Strategy for Solving Related Rates Problems #4
2:50
Strategy for Solving Related Rates Problems #5
3:38
Example I: Radius of a Balloon
5:15
Example II: Ladder
12:52
Example III: Water Tank
19:08
Example IV: Distance between Two Cars
29:27
Example V: Line-of-Sight
36:20
More Related Rates Examples

37m 17s

Intro
0:00
Example I: Shadow
0:14
Example II: Particle
4:45
Example III: Water Level
10:28
Example IV: Clock
20:47
Example V: Distance between a House and a Plane
29:11
Maximum & Minimum Values of a Function

40m 44s

Intro
0:00
Maximum & Minimum Values of a Function, Part 1
0:23
Absolute Maximum
2:20
Absolute Minimum
2:52
Local Maximum
3:38
Local Minimum
4:26
Maximum & Minimum Values of a Function, Part 2
6:11
Function with Absolute Minimum but No Absolute Max, Local Max, and Local Min
7:18
Function with Local Max & Min but No Absolute Max & Min
8:48
Formal Definitions
10:43
Absolute Maximum
11:18
Absolute Minimum
12:57
Local Maximum
14:37
Local Minimum
16:25
Extreme Value Theorem
18:08
Theorem: f'(c) = 0
24:40
Critical Number (Critical Value)
26:14
Procedure for Finding the Critical Values of f(x)
28:32
Example I: Find the Critical Values of f(x) x + sinx
29:51
Example II: What are the Absolute Max & Absolute Minimum of f(x) = x + 4 sinx on [0,2π]
35:31
Example Problems for Max & Min

40m 44s

Intro
0:00
Example I: Identify Absolute and Local Max & Min on the Following Graph
0:11
Example II: Sketch the Graph of a Continuous Function
3:11
Example III: Sketch the Following Graphs
4:40
Example IV: Find the Critical Values of f (x) = 3x⁴ - 7x³ + 4x²
6:13
Example V: Find the Critical Values of f(x) = |2x - 5|
8:42
Example VI: Find the Critical Values
11:42
Example VII: Find the Critical Values f(x) = cos²(2x) on [0,2π]
16:57
Example VIII: Find the Absolute Max & Min f(x) = 2sinx + 2cos x on [0,(π/3)]
20:08
Example IX: Find the Absolute Max & Min f(x) = (ln(2x)) / x on [1,3]
24:39
The Mean Value Theorem

25m 54s

Intro
0:00
Rolle's Theorem
0:08
Rolle's Theorem: If & Then
0:09
Rolle's Theorem: Geometrically
2:06
There May Be More than 1 c Such That f'( c ) = 0
3:30
Example I: Rolle's Theorem
4:58
The Mean Value Theorem
9:12
The Mean Value Theorem: If & Then
9:13
The Mean Value Theorem: Geometrically
11:07
Example II: Mean Value Theorem
13:43
Example III: Mean Value Theorem
21:19
Using Derivatives to Graph Functions, Part I

25m 54s

Intro
0:00
Using Derivatives to Graph Functions, Part I
0:12
Increasing/ Decreasing Test
0:13
Example I: Find the Intervals Over Which the Function is Increasing & Decreasing
3:26
Example II: Find the Local Maxima & Minima of the Function
19:18
Example III: Find the Local Maxima & Minima of the Function
31:39
Using Derivatives to Graph Functions, Part II

44m 58s

Intro
0:00
Using Derivatives to Graph Functions, Part II
0:13
Concave Up & Concave Down
0:14
What Does This Mean in Terms of the Derivative?
6:14
Point of Inflection
8:52
Example I: Graph the Function
13:18
Example II: Function x⁴ - 5x²
19:03
Intervals of Increase & Decrease
19:04
Local Maxes and Mins
25:01
Intervals of Concavity & X-Values for the Points of Inflection
29:18
Intervals of Concavity & Y-Values for the Points of Inflection
34:18
Graphing the Function
40:52
Example Problems I

49m 19s

Intro
0:00
Example I: Intervals, Local Maxes & Mins
0:26
Example II: Intervals, Local Maxes & Mins
5:05
Example III: Intervals, Local Maxes & Mins, and Inflection Points
13:40
Example IV: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity
23:02
Example V: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity
34:36
Example Problems III

59m 1s

Intro
0:00
Example I: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
0:11
Example II: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
21:24
Example III: Cubic Equation f(x) = Ax³ + Bx² + Cx + D
37:56
Example IV: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
46:19
L'Hospital's Rule

30m 9s

Intro
0:00
L'Hospital's Rule
0:19
Indeterminate Forms
0:20
L'Hospital's Rule
3:38
Example I: Evaluate the Following Limit Using L'Hospital's Rule
8:50
Example II: Evaluate the Following Limit Using L'Hospital's Rule
10:30
Indeterminate Products
11:54
Indeterminate Products
11:55
Example III: L'Hospital's Rule & Indeterminate Products
13:57
Indeterminate Differences
17:00
Indeterminate Differences
17:01
Example IV: L'Hospital's Rule & Indeterminate Differences
18:57
Indeterminate Powers
22:20
Indeterminate Powers
22:21
Example V: L'Hospital's Rule & Indeterminate Powers
25:13
Example Problems for L'Hospital's Rule

38m 14s

Intro
0:00
Example I: Evaluate the Following Limit
0:17
Example II: Evaluate the Following Limit
2:45
Example III: Evaluate the Following Limit
6:54
Example IV: Evaluate the Following Limit
8:43
Example V: Evaluate the Following Limit
11:01
Example VI: Evaluate the Following Limit
14:48
Example VII: Evaluate the Following Limit
17:49
Example VIII: Evaluate the Following Limit
20:37
Example IX: Evaluate the Following Limit
25:16
Example X: Evaluate the Following Limit
32:44
Optimization Problems I

49m 59s

Intro
0:00
Example I: Find the Dimensions of the Box that Gives the Greatest Volume
1:23
Fundamentals of Optimization Problems
18:08
Fundamental #1
18:33
Fundamental #2
19:09
Fundamental #3
19:19
Fundamental #4
20:59
Fundamental #5
21:55
Fundamental #6
23:44
Example II: Demonstrate that of All Rectangles with a Given Perimeter, the One with the Largest Area is a Square
24:36
Example III: Find the Points on the Ellipse 9x² + y² = 9 Farthest Away from the Point (1,0)
35:13
Example IV: Find the Dimensions of the Rectangle of Largest Area that can be Inscribed in a Circle of Given Radius R
43:10
Optimization Problems II

55m 10s

Intro
0:00
Example I: Optimization Problem
0:13
Example II: Optimization Problem
17:34
Example III: Optimization Problem
35:06
Example IV: Revenue, Cost, and Profit
43:22
Newton's Method

30m 22s

Intro
0:00
Newton's Method
0:45
Newton's Method
0:46
Example I: Find x2 and x3
13:18
Example II: Use Newton's Method to Approximate
15:48
Example III: Find the Root of the Following Equation to 6 Decimal Places
19:57
Example IV: Use Newton's Method to Find the Coordinates of the Inflection Point
23:11
IV. Integrals
Antiderivatives

55m 26s

Intro
0:00
Antiderivatives
0:23
Definition of an Antiderivative
0:24
Antiderivative Theorem
7:58
Function & Antiderivative
12:10
x^n
12:30
1/x
13:00
e^x
13:08
cos x
13:18
sin x
14:01
sec² x
14:11
secxtanx
14:18
1/√(1-x²)
14:26
1/(1+x²)
14:36
-1/√(1-x²)
14:45
Example I: Find the Most General Antiderivative for the Following Functions
15:07
Function 1: f(x) = x³ -6x² + 11x - 9
15:42
Function 2: f(x) = 14√(x) - 27 4√x
19:12
Function 3: (fx) = cos x - 14 sinx
20:53
Function 4: f(x) = (x⁵+2√x )/( x^(4/3) )
22:10
Function 5: f(x) = (3e^x) - 2/(1+x²)
25:42
Example II: Given the Following, Find the Original Function f(x)
26:37
Function 1: f'(x) = 5x³ - 14x + 24, f(2) = 40
27:55
Function 2: f'(x) 3 sinx + sec²x, f(π/6) = 5
30:34
Function 3: f''(x) = 8x - cos x, f(1.5) = 12.7, f'(1.5) = 4.2
32:54
Function 4: f''(x) = 5/(√x), f(2) 15, f'(2) = 7
37:54
Example III: Falling Object
41:58
Problem 1: Find an Equation for the Height of the Ball after t Seconds
42:48
Problem 2: How Long Will It Take for the Ball to Strike the Ground?
48:30
Problem 3: What is the Velocity of the Ball as it Hits the Ground?
49:52
Problem 4: Initial Velocity of 6 m/s, How Long Does It Take to Reach the Ground?
50:46
The Area Under a Curve

51m 3s

Intro
0:00
The Area Under a Curve
0:13
Approximate Using Rectangles
0:14
Let's Do This Again, Using 4 Different Rectangles
9:40
Approximate with Rectangles
16:10
Left Endpoint
18:08
Right Endpoint
25:34
Left Endpoint vs. Right Endpoint
30:58
Number of Rectangles
34:08
True Area
37:36
True Area
37:37
Sigma Notation & Limits
43:32
When You Have to Explicitly Solve Something
47:56
Example Problems for Area Under a Curve

33m 7s

Intro
0:00
Example I: Using Left Endpoint & Right Endpoint to Approximate Area Under a Curve
0:10
Example II: Using 5 Rectangles, Approximate the Area Under the Curve
11:32
Example III: Find the True Area by Evaluating the Limit Expression
16:07
Example IV: Find the True Area by Evaluating the Limit Expression
24:52
The Definite Integral

43m 19s

Intro
0:00
The Definite Integral
0:08
Definition to Find the Area of a Curve
0:09
Definition of the Definite Integral
4:08
Symbol for Definite Integral
8:45
Regions Below the x-axis
15:18
Associating Definite Integral to a Function
19:38
Integrable Function
27:20
Evaluating the Definite Integral
29:26
Evaluating the Definite Integral
29:27
Properties of the Definite Integral
35:24
Properties of the Definite Integral
35:25
Example Problems for The Definite Integral

32m 14s

Intro
0:00
Example I: Approximate the Following Definite Integral Using Midpoints & Sub-intervals
0:11
Example II: Express the Following Limit as a Definite Integral
5:28
Example III: Evaluate the Following Definite Integral Using the Definition
6:28
Example IV: Evaluate the Following Integral Using the Definition
17:06
Example V: Evaluate the Following Definite Integral by Using Areas
25:41
Example VI: Definite Integral
30:36
The Fundamental Theorem of Calculus

24m 17s

Intro
0:00
The Fundamental Theorem of Calculus
0:17
Evaluating an Integral
0:18
Lim as x → ∞
12:19
Taking the Derivative
14:06
Differentiation & Integration are Inverse Processes
15:04
1st Fundamental Theorem of Calculus
20:08
1st Fundamental Theorem of Calculus
20:09
2nd Fundamental Theorem of Calculus
22:30
2nd Fundamental Theorem of Calculus
22:31
Example Problems for the Fundamental Theorem

25m 21s

Intro
0:00
Example I: Find the Derivative of the Following Function
0:17
Example II: Find the Derivative of the Following Function
1:40
Example III: Find the Derivative of the Following Function
2:32
Example IV: Find the Derivative of the Following Function
5:55
Example V: Evaluate the Following Integral
7:13
Example VI: Evaluate the Following Integral
9:46
Example VII: Evaluate the Following Integral
12:49
Example VIII: Evaluate the Following Integral
13:53
Example IX: Evaluate the Following Graph
15:24
Local Maxs and Mins for g(x)
15:25
Where Does g(x) Achieve Its Absolute Max on [0,8]
20:54
On What Intervals is g(x) Concave Up/Down?
22:20
Sketch a Graph of g(x)
24:34
More Example Problems, Including Net Change Applications

34m 22s

Intro
0:00
Example I: Evaluate the Following Indefinite Integral
0:10
Example II: Evaluate the Following Definite Integral
0:59
Example III: Evaluate the Following Integral
2:59
Example IV: Velocity Function
7:46
Part A: Net Displacement
7:47
Part B: Total Distance Travelled
13:15
Example V: Linear Density Function
20:56
Example VI: Acceleration Function
25:10
Part A: Velocity Function at Time t
25:11
Part B: Total Distance Travelled During the Time Interval
28:38
Solving Integrals by Substitution

27m 20s

Intro
0:00
Table of Integrals
0:35
Example I: Evaluate the Following Indefinite Integral
2:02
Example II: Evaluate the Following Indefinite Integral
7:27
Example IIII: Evaluate the Following Indefinite Integral
10:57
Example IV: Evaluate the Following Indefinite Integral
12:33
Example V: Evaluate the Following
14:28
Example VI: Evaluate the Following
16:00
Example VII: Evaluate the Following
19:01
Example VIII: Evaluate the Following
21:49
Example IX: Evaluate the Following
24:34
V. Applications of Integration
Areas Between Curves

34m 56s

Intro
0:00
Areas Between Two Curves: Function of x
0:08
Graph 1: Area Between f(x) & g(x)
0:09
Graph 2: Area Between f(x) & g(x)
4:07
Is It Possible to Write as a Single Integral?
8:20
Area Between the Curves on [a,b]
9:24
Absolute Value
10:32
Formula for Areas Between Two Curves: Top Function - Bottom Function
17:03
Areas Between Curves: Function of y
17:49
What if We are Given Functions of y?
17:50
Formula for Areas Between Two Curves: Right Function - Left Function
21:48
Finding a & b
22:32
Example Problems for Areas Between Curves

42m 55s

Intro
0:00
Instructions for the Example Problems
0:10
Example I: y = 7x - x² and y=x
0:37
Example II: x=y²-3, x=e^((1/2)y), y=-1, and y=2
6:25
Example III: y=(1/x), y=(1/x³), and x=4
12:25
Example IV: 15-2x² and y=x²-5
15:52
Example V: x=(1/8)y³ and x=6-y²
20:20
Example VI: y=cos x, y=sin(2x), [0,π/2]
24:34
Example VII: y=2x², y=10x², 7x+2y=10
29:51
Example VIII: Velocity vs. Time
33:23
Part A: At 2.187 Minutes, Which care is Further Ahead?
33:24
Part B: If We Shaded the Region between the Graphs from t=0 to t=2.187, What Would This Shaded Area Represent?
36:32
Part C: At 4 Minutes Which Car is Ahead?
37:11
Part D: At What Time Will the Cars be Side by Side?
37:50
Volumes I: Slices

34m 15s

Intro
0:00
Volumes I: Slices
0:18
Rotate the Graph of y=√x about the x-axis
0:19
How can I use Integration to Find the Volume?
3:16
Slice the Solid Like a Loaf of Bread
5:06
Volumes Definition
8:56
Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
12:18
Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
19:05
Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
25:28
Volumes II: Volumes by Washers

51m 43s

Intro
0:00
Volumes II: Volumes by Washers
0:11
Rotating Region Bounded by y=x³ & y=x around the x-axis
0:12
Equation for Volumes by Washer
11:14
Process for Solving Volumes by Washer
13:40
Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
15:58
Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
25:07
Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
34:20
Example IV: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
44:05
Volumes III: Solids That Are Not Solids-of-Revolution

49m 36s

Intro
0:00
Solids That Are Not Solids-of-Revolution
0:11
Cross-Section Area Review
0:12
Cross-Sections That Are Not Solids-of-Revolution
7:36
Example I: Find the Volume of a Pyramid Whose Base is a Square of Side-length S, and Whose Height is H
10:54
Example II: Find the Volume of a Solid Whose Cross-sectional Areas Perpendicular to the Base are Equilateral Triangles
20:39
Example III: Find the Volume of a Pyramid Whose Base is an Equilateral Triangle of Side-Length A, and Whose Height is H
29:27
Example IV: Find the Volume of a Solid Whose Base is Given by the Equation 16x² + 4y² = 64
36:47
Example V: Find the Volume of a Solid Whose Base is the Region Bounded by the Functions y=3-x² and the x-axis
46:13
Volumes IV: Volumes By Cylindrical Shells

50m 2s

Intro
0:00
Volumes by Cylindrical Shells
0:11
Find the Volume of the Following Region
0:12
Volumes by Cylindrical Shells: Integrating Along x
14:12
Volumes by Cylindrical Shells: Integrating Along y
14:40
Volumes by Cylindrical Shells Formulas
16:22
Example I: Using the Method of Cylindrical Shells, Find the Volume of the Solid
18:33
Example II: Using the Method of Cylindrical Shells, Find the Volume of the Solid
25:57
Example III: Using the Method of Cylindrical Shells, Find the Volume of the Solid
31:38
Example IV: Using the Method of Cylindrical Shells, Find the Volume of the Solid
38:44
Example V: Using the Method of Cylindrical Shells, Find the Volume of the Solid
44:03
The Average Value of a Function

32m 13s

Intro
0:00
The Average Value of a Function
0:07
Average Value of f(x)
0:08
What if The Domain of f(x) is Not Finite?
2:23
Let's Calculate Average Value for f(x) = x² [2,5]
4:46
Mean Value Theorem for Integrate
9:25
Example I: Find the Average Value of the Given Function Over the Given Interval
14:06
Example II: Find the Average Value of the Given Function Over the Given Interval
18:25
Example III: Find the Number A Such that the Average Value of the Function f(x) = -4x² + 8x + 4 Equals 2 Over the Interval [-1,A]
24:04
Example IV: Find the Average Density of a Rod
27:47
VI. Techniques of Integration
Integration by Parts

50m 32s

Intro
0:00
Integration by Parts
0:08
The Product Rule for Differentiation
0:09
Integrating Both Sides Retains the Equality
0:52
Differential Notation
2:24
Example I: ∫ x cos x dx
5:41
Example II: ∫ x² sin(2x)dx
12:01
Example III: ∫ (e^x) cos x dx
18:19
Example IV: ∫ (sin^-1) (x) dx
23:42
Example V: ∫₁⁵ (lnx)² dx
28:25
Summary
32:31
Tabular Integration
35:08
Case 1
35:52
Example: ∫x³sinx dx
36:39
Case 2
40:28
Example: ∫e^(2x) sin 3x
41:14
Trigonometric Integrals I

24m 50s

Intro
0:00
Example I: ∫ sin³ (x) dx
1:36
Example II: ∫ cos⁵(x)sin²(x)dx
4:36
Example III: ∫ sin⁴(x)dx
9:23
Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sin^m) (x) (cos^p) (x) dx
15:59
#1: Power of sin is Odd
16:00
#2: Power of cos is Odd
16:41
#3: Powers of Both sin and cos are Odd
16:55
#4: Powers of Both sin and cos are Even
17:10
Example IV: ∫ tan⁴ (x) sec⁴ (x) dx
17:34
Example V: ∫ sec⁹(x) tan³(x) dx
20:55
Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sec^m) (x) (tan^p) (x) dx
23:31
#1: Power of sec is Odd
23:32
#2: Power of tan is Odd
24:04
#3: Powers of sec is Odd and/or Power of tan is Even
24:18
Trigonometric Integrals II

22m 12s

Intro
0:00
Trigonometric Integrals II
0:09
Recall: ∫tanx dx
0:10
Let's Find ∫secx dx
3:23
Example I: ∫ tan⁵ (x) dx
6:23
Example II: ∫ sec⁵ (x) dx
11:41
Summary: How to Deal with Integrals of Different Types
19:04
Identities to Deal with Integrals of Different Types
19:05
Example III: ∫cos(5x)sin(9x)dx
19:57
More Example Problems for Trigonometric Integrals

17m 22s

Intro
0:00
Example I: ∫sin²(x)cos⁷(x)dx
0:14
Example II: ∫x sin²(x) dx
3:56
Example III: ∫csc⁴ (x/5)dx
8:39
Example IV: ∫( (1-tan²x)/(sec²x) ) dx
11:17
Example V: ∫ 1 / (sinx-1) dx
13:19
Integration by Partial Fractions I

55m 12s

Intro
0:00
Integration by Partial Fractions I
0:11
Recall the Idea of Finding a Common Denominator
0:12
Decomposing a Rational Function to Its Partial Fractions
4:10
2 Types of Rational Function: Improper & Proper
5:16
Improper Rational Function
7:26
Improper Rational Function
7:27
Proper Rational Function
11:16
Proper Rational Function & Partial Fractions
11:17
Linear Factors
14:04
Irreducible Quadratic Factors
15:02
Case 1: G(x) is a Product of Distinct Linear Factors
17:10
Example I: Integration by Partial Fractions
20:33
Case 2: D(x) is a Product of Linear Factors
40:58
Example II: Integration by Partial Fractions
44:41
Integration by Partial Fractions II

42m 57s

Intro
0:00
Case 3: D(x) Contains Irreducible Factors
0:09
Example I: Integration by Partial Fractions
5:19
Example II: Integration by Partial Fractions
16:22
Case 4: D(x) has Repeated Irreducible Quadratic Factors
27:30
Example III: Integration by Partial Fractions
30:19
VII. Differential Equations
Introduction to Differential Equations

46m 37s

Intro
0:00
Introduction to Differential Equations
0:09
Overview
0:10
Differential Equations Involving Derivatives of y(x)
2:08
Differential Equations Involving Derivatives of y(x) and Function of y(x)
3:23
Equations for an Unknown Number
6:28
What are These Differential Equations Saying?
10:30
Verifying that a Function is a Solution of the Differential Equation
13:00
Verifying that a Function is a Solution of the Differential Equation
13:01
Verify that y(x) = 4e^x + 3x² + 6x + e^π is a Solution of this Differential Equation
17:20
General Solution
22:00
Particular Solution
24:36
Initial Value Problem
27:42
Example I: Verify that a Family of Functions is a Solution of the Differential Equation
32:24
Example II: For What Values of K Does the Function Satisfy the Differential Equation
36:07
Example III: Verify the Solution and Solve the Initial Value Problem
39:47
Separation of Variables

28m 8s

Intro
0:00
Separation of Variables
0:28
Separation of Variables
0:29
Example I: Solve the Following g Initial Value Problem
8:29
Example II: Solve the Following g Initial Value Problem
13:46
Example III: Find an Equation of the Curve
18:48
Population Growth: The Standard & Logistic Equations

51m 7s

Intro
0:00
Standard Growth Model
0:30
Definition of the Standard/Natural Growth Model
0:31
Initial Conditions
8:00
The General Solution
9:16
Example I: Standard Growth Model
10:45
Logistic Growth Model
18:33
Logistic Growth Model
18:34
Solving the Initial Value Problem
25:21
What Happens When t → ∞
36:42
Example II: Solve the Following g Initial Value Problem
41:50
Relative Growth Rate
46:56
Relative Growth Rate
46:57
Relative Growth Rate Version for the Standard model
49:04
Slope Fields

24m 37s

Intro
0:00
Slope Fields
0:35
Slope Fields
0:36
Graphing the Slope Fields, Part 1
11:12
Graphing the Slope Fields, Part 2
15:37
Graphing the Slope Fields, Part 3
17:25
Steps to Solving Slope Field Problems
20:24
Example I: Draw or Generate the Slope Field of the Differential Equation y'=x cos y
22:38
VIII. AP Practic Exam
AP Practice Exam: Section 1, Part A No Calculator

45m 29s

Intro
0:00
Exam Link
0:10
Problem #1
1:26
Problem #2
2:52
Problem #3
4:42
Problem #4
7:03
Problem #5
10:01
Problem #6
13:49
Problem #7
15:16
Problem #8
19:06
Problem #9
23:10
Problem #10
28:10
Problem #11
31:30
Problem #12
33:53
Problem #13
37:45
Problem #14
41:17
AP Practice Exam: Section 1, Part A No Calculator, cont.

41m 55s

Intro
0:00
Problem #15
0:22
Problem #16
3:10
Problem #17
5:30
Problem #18
8:03
Problem #19
9:53
Problem #20
14:51
Problem #21
17:30
Problem #22
22:12
Problem #23
25:48
Problem #24
29:57
Problem #25
33:35
Problem #26
35:57
Problem #27
37:57
Problem #28
40:04
AP Practice Exam: Section I, Part B Calculator Allowed

58m 47s

Intro
0:00
Problem #1
1:22
Problem #2
4:55
Problem #3
10:49
Problem #4
13:05
Problem #5
14:54
Problem #6
17:25
Problem #7
18:39
Problem #8
20:27
Problem #9
26:48
Problem #10
28:23
Problem #11
34:03
Problem #12
36:25
Problem #13
39:52
Problem #14
43:12
Problem #15
47:18
Problem #16
50:41
Problem #17
56:38
AP Practice Exam: Section II, Part A Calculator Allowed

25m 40s

Intro
0:00
Problem #1: Part A
1:14
Problem #1: Part B
4:46
Problem #1: Part C
8:00
Problem #2: Part A
12:24
Problem #2: Part B
16:51
Problem #2: Part C
17:17
Problem #3: Part A
18:16
Problem #3: Part B
19:54
Problem #3: Part C
21:44
Problem #3: Part D
22:57
AP Practice Exam: Section II, Part B No Calculator

31m 20s

Intro
0:00
Problem #4: Part A
1:35
Problem #4: Part B
5:54
Problem #4: Part C
8:50
Problem #4: Part D
9:40
Problem #5: Part A
11:26
Problem #5: Part B
13:11
Problem #5: Part C
15:07
Problem #5: Part D
19:57
Problem #6: Part A
22:01
Problem #6: Part B
25:34
Problem #6: Part C
28:54
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Example Problems for Area Under a Curve

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Example I: Using Left Endpoint & Right Endpoint to Approximate Area Under a Curve 0:10
  • Example II: Using 5 Rectangles, Approximate the Area Under the Curve 11:32
  • Example III: Find the True Area by Evaluating the Limit Expression 16:07
  • Example IV: Find the True Area by Evaluating the Limit Expression 24:52

Transcription: Example Problems for Area Under a Curve

Hello, welcome back to www.educator.com, welcome back to AP Calculus.0000

Today, we are going to do some example problems for the area under a curve.0004

Let us get started.0009

Use the graphs on the following pages to find left endpoint,0012

right endpoint approximations for the area under the curve from 0 to 12.0018

First use n = 6, we are going to use 6 rectangles then do the same for n = 12.0022

Draw all the appropriate rectangles for each case.0030

There are 4 copies of the graph, one for each case, do the work, write on the graph, find areas, things like that.0033

Let us get started.0041

For here, we have n = 6 and we are going to do left endpoints.0043

We know the a is equal to 0, we know that b is equal to 12.0052

We have our 0 here and we have our 12 over here.0057

We know that Δx, that is going to be b - a/n, which is 12 - 0/ 6, which is equal to 2.0063

Our Δx is going to equal 2.0076

Left endpoints, left endpoint, go up to the graph, draw your rectangle.0080

This is the endpoint, go up to the graph, draw your rectangle.0090

Go up to the graph, draw your rectangle.0096

Go up to the graph, go to the right to the next x value, draw your rectangle.0103

Go up to the graph to your next x point, draw your rectangle.0108

Go up to the graph, rectangle.0112

There you go, 1, 2, 3, 4, 5, 6, Δx is 2.0116

Therefore, our area, left, 6 rectangles, is equal to 2 which is the Δx.0124

It is going to be f of this + f of that + f of this + f of that + f of this + f of that.0133

1, 2, 3, 4, 5, 6, read it right off the graph.0141

This x value is going to be 9.0147

I read it right off the graph, it is 8.8.0153

It is going to be 8.4, the next one is 7.6 + 6.4.0158

My last one is going to be 5, there we go.0168

I get a value of 90.4 using these.0173

My left endpoint approximation using 6 rectangles gives me 90.4.0181

We are going to do again n = 6.0190

We are going to do right endpoints.0195

Once again, a is equal to 0, b is equal to 12.0199

It is the area I’m looking for, which means my Δx is going to be 12 - 0/6.0206

Again, we are looking at 2 right endpoints.0213

I start from the right, go up to the graph, and I go to the left, until I hit the next point.0217

It is that one, 12, this is 2.0225

That is what this means, the base of my rectangles have a length of 2.0229

From here, I go up to the graph, and I go to the left to the next x endpoint.0232

That is my second rectangle.0238

I go up to the graph, go to the left, that is my 3rd rectangle.0241

Up the graph, to the left, that is my 4th rectangle.0247

Up to the graph, to the left, that is my 5th rectangle.0253

Up to the graph, left, that is my 6th rectangle.0257

Let us go ahead and do 1, 2, 3, 4, 5, 6.0266

You can number them anyway you want.0270

You can go from the left to the right, it does not really matter.0271

My area with right endpoints using 6 rectangles is 2 Δx ×, we are starting over here, or I can start over here.0275

It just depends on how you want to do it.0288

We are using right endpoints, I just going to go ahead and start over here.0293

When I do that, it is going to be 3.2 + 5 + 6.4 + 7.6 + 8.4 + 8.8.0299

3.2, 5, 6.4, 7.6, 8.4, 8.8.0321

When I do that, I get 78.8.0331

Now we are going to do 12 rectangles.0342

This time, n = 12.0345

Once again, Δx is equal to 12 - 0/12.0348

Now the Δx is 1, I have 12 rectangles.0355

We are going to do the left endpoints.0359

From the left endpoint 0, right endpoint is 12.0365

Up, go across, that is one rectangle.0371

Up, go across, 2nd rectangle.0376

Up, go across, 3rd rectangle.0379

Up to the graph, go across, 4th rectangle.0383

Up, go across, up to the graph, go across.0386

Up to the graph, go across, up to the graph, go across.0393

Up to the graph, go across, up to the graph, go across.0398

Up to the graph, go across, up to the graph, go across.0404

There you go, the points 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.0409

Those are the points that I'm going to be looking at the area.0419

Using left endpoints 12, it is going to equal, Δx is 1.0422

That height, that height, that height, 9.0428

I have got 9 + 8.9 + 8.8 + 8.6 + 8.4 + 8 + 7.6 + 7 + 6.4 + 5.7 + 5 + 4.2.0440

I get a value of 87.6.0468

These are my f values.0475

Those numbers that I have read off the graph are these right here.0481

Left endpoints, right endpoints, n = 12, Δx = 12 - 0/12 = 1.0485

Starting from the right, starting from here, go up to the graph, go across, that is my first rectangle.0504

Go up, go across, 2nd rectangle.0512

Go up, go across, 3rd rectangle.0516

Go up, go across, 4th rectangle.0520

Go up to the graph, go across, 5th rectangle.0524

Go up to the graph, go across, 6th rectangle, 7th rectangle, 8th rectangle, 9th rectangle, 10th rectangle, 11th rectangle, 12th rectangle.0528

4, 5, 6, 7, 8, 9, 10, 11, 12.0554

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.0564

The heights are going to be 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.0570

The area that you get is area using right endpoints.0581

12 rectangles is 1 × we are going to have 3.2 + 4.2 + 5 + 5.70585

+ 6.4 + 7 + 7.6 + 8 + 8.4 + 8.6 + 8.8 + 8.9.0600

We are going to get a value of 81. 8.0617

Here is what we have, a left 6 was 90.4.0621

A left 12 was 87.6, a right 6 was 78.8.0632

A right 12 was 81.8, this is dropping as I went from 6 to 12.0651

As I made more rectangles, smaller rectangles, this one went up.0659

There is going to be some value that they are going to hit, some a true value.0665

The true value actually happens to be 84.96.0671

You will find out how to get that, in a couple of lessons from now.0675

This drops, this rises, lower sum comes up, upper sum comes down.0681

There is some point where they meet, that point is the true area.0686

It turns out to be 84.96.0689

Using 5 rectangles, approximate the area under the graph of f(x) = x² + 2, from x = -2 to 3 using midpoints.0694

That is use the point midway between the left and right endpoints of a given approximating rectangle.0704

We have got n = 5, we are using midpoints Δx = b - a/ n which is 3, - and - 2/ 5 which is 1.0711

The base length of the rectangle is equal to 1.0736

We are using midpoints.0740

The points I’m interested are, this is one rectangle right here, midpoint between the x values.0747

This is going to be a rectangle but I’m using the midpoint of the rectangle, not the left endpoint, not the right endpoint.0755

I'm using midpoints, that is another point, that is another point, that is another point, that is another point.0762

These x values, that is going to be my x.0768

x sub 1,x sub 2, x sub 4, x sub 5.0775

Now from those points, I go up to the graph and I go to the left endpoint, to the right, the right endpoint, that is my rectangle.0780

I’m going to go ahead and draw this as a dotted line.0794

From whatever point that I choose, that is what I go up to the graph.0802

That is going to be my f value.0807

From this midpoint, I go up to the graph, that is going to be my f value.0808

From there, I go to the right endpoint, to the left endpoint which is 1.0813

1, that is the width of my rectangle, I just happen to be taking points in between.0819

From the midpoint, I go up to the graph, I go to the right, to one x value, to the left x value.0824

That is my other rectangle, 1 rectangle, 2 rectangle, 3 rectangle, midpoint.0831

I go up to that point, I go to the right.0839

To one x, I go to the left to one x, that is my rectangle.0847

Midpoint, I go up there, I go to the right, to one, to left, to one, that is my other rectangle.0852

1, 2, 3, 4, 5, and these are my rectangles.0864

My area, midpoints using 5 rectangles is going to be 1 ×, it is going to be f(-1.5) + f(-0.5)0870

+ f(0.5) + f(1.5) + f(2.5).0889

These are the points that I'm going to take the f value of, that I’m going to put into my function x² + 2 to get the y value.0902

The area of the midpoints 5 is equal to 1 ×, it is going to be 4.25 + 2.25 + 2.25 + 4.25 + 7.0625.0914

The area is going to equal 20.0625, the approximate area.0931

I hope that made sense.0940

I have my points but I’m using midpoints.0941

From my rectangle, from that midpoint, that is where I go up to the graph and0945

I go left and right to the x value below it, the x value above it.0949

This is the height of my rectangle now.0954

Not left endpoint, up to the graph, or right endpoint, up to the graph, but a midpoint up to the graph, and go left and right.0956

Let us see, given the function f(x) = 3√x from 0 to 27, write an expression for the true area under the graph using limits in sigma notation.0969

Let us find Δx first, let me go to blue.0981

I have got Δx = b - a/ n, that is going to be 27 – 0/ n, which is going to be 27/n, that is my Δx.0987

My area is the limit as n goes to infinity of the sum i = 1 to n of, we said Δx.0999

The equation was the limit as n goes to infinity of the sum of Δx × f(x) sub i.1017

Δx is 27/ n.1031

F is this, so it is going to be 3√x sub i.1038

We need to find what x sub i is, in terms of n.1047

We know what the x sub i are going to be, we have a beginning point, we have an ending point.1055

We actually can find some expression for x sub I, to put into this.1061

Let us list them, we have x sub 1, that is just 0, that is the left endpoint.1067

x sub 2, that is 0 + Δx, that is going to be 27/n.1074

x sub 3, that is just x sub 2 which is 27/n + another Δx, that is going to be 54/n.1083

x sub 4 equal, it is going to be 54/n + another Δx, that is going to be 81/n.1092

x sub i is going to be some, we are going to be looking for some function of i/n.1103

Let me not write this just yet, I'm looking for some sort of a pattern I can find here.1110

Some relationship between, I see that there are n’s in the denominator and I see a 27, 54, and 81.1116

I need to find a relationship between 27, 2, 54, 3, 81, and 4.1124

In other words, is there a relationship between the I and some f(i) which will give me these numbers.1129

I'm looking for an expression that I can substitute.1143

I’m looking for an x sub i that I can substitute into here, that I can substitute for x sub i.1164

I need some relation between i and n.1179

In other words, I'm looking for some x sub i which is going to be some function of i/n.1192

Based on this pattern that I elucidated from the first x sub 1, the 2nd point, the 3rd point, the 4th point.1199

I saw this pattern, I’m looking for something here.1208

I have got it, this is 2 and this is 27.1214

This is 3 and this is 54.1219

This is 4 and this is 81.1222

27 is 1 ×, if i is 2, then this is 2 - 1 × 27.1224

If i is 3, then this is 3 - 1 which is 2 × 27.1237

If i is 4, 81 is equal to 3 × 27.1243

Therefore, my x sub i is going to equal i - 1 × 27/n.1249

This relationship, as i runs 1, 2, 3, 4, 5, 6, 7, gives me these numbers.1260

Now I can put this into here and what I end up with is the following.1272

A = the limit as n goes to infinity, the sum as i runs from 1 to n of 27/n × 3√27 × i - 1/n.1281

Those are perfectly acceptable, I can absolutely leave it like this.1305

However, there is another way to simplify this thing, to simplify the function.1311

I can adjust the index as follows.1330

Now I had x sub 1 = 0, x sub 2 = 27/n, I had x sub 3 = 54/n, I had x sub 4 = 81/n.1340

Therefore, I had x sub i is equal to i - 1 × the 27/n.1357

I can do this, we can do, we can set a equal to 0.1365

And then, we can call x sub 1 27/n, x sub 2 54/n, x sub 3 is 81/n.1378

Therefore, x sub i is just 27i/ n.1390

It does not have that i - 1 quality.1396

This is just 1, 27, 2, 2 × 27 is 54, 3, 3 × 27 is 81.1398

i × 27, it makes it a little bit easier.1405

All I have done is instead of calling x1 0, I just shifted the index down.1410

I just called a = 0, I have called x1 the 27/n, I shifted the index.1415

I end up with a is equal to the limit as n goes to infinity, the sum from 1 to n of 27/n × 3√27 i/ n.1424

It is a lot simpler and I can actually pull out the 3√27 here.1443

I can write a = the limit as n goes to infinity of the sum i = 1 to n.1447

3√27 is 3, 3 × 27 is 81.1457

We get 81/n × 3√i/n.1460

When I actually evaluate the sum which I will do in the next problem, a version of that,1472

I just pull out all the n, all that I'm left with under the summation sign is an i.1476

Everything else comes out as a constant.1481

It is really nice.1484

That is our final answer, or this one, either one is fine.1486

Given the function f(x) = 4x³ from 0 to 2, find the true area by actually evaluating the limit expression that you obtain.1495

Use the following fact about the sum of cubes in the first n integers.1504

When I end up with a sum of cubes, 1³ + 2³ + 3³, all the way to whatever number n³,1508

I have a closed form expression for that, that is this.1517

If I have a sum that I can express as this, I can express in sigma notation, I can replace that sigma notation with this.1520

That is what this is saying.1528

Let us go ahead and do this problem.1530

Let us start with our Δx, I’m going to start over here.1533

Δx = b - a/ n, it is just going to be 2/n.1536

I know, therefore, that my area is going to equal the limit as n goes to infinity of the sum1547

as i goes from 1 to n of Δx, which is my 2/n × 4 × x sub i³.1558

That is my definition.1571

My question is, what can we put for this x sub i in there?1575

Let us see what we have got.1587

Let us come over here.1589

I think I’m going to do, I’m going to do what I did last time.1595

I’m going to set a equal to 0, and I’m going to set x sub 1 equal to, it is 0 + Δx which is 2/n.1599

x sub 2 = 2/n + 2/n which is 4/n.1612

x sub 3 = 4/n + 2/n which is 6/n.1619

I got my relationship 1, 2, 2, 4, 3, 6, x sub i is equal to 2 sub i/n.1624

A is equal to the limit as n goes to infinity, i goes from 1 to n of 2/n × 4 x sub i³.1649

x sub i is this, 4 × 2i/ n³.1665

This is going to equal the limit as n goes to infinity of the sum i = 1 to n.1676

2 × 4 is 8, this is 8/n × 8 i³/ n³.1686

We are just doing everything that we need to do.1698

This is a function of i, the index is what the variable is under a summation sign.1707

8 × 8 is 64, n × n³ = n⁴.1714

I’m just going to pull all of that and take it outside of the summation sign, which I can actually do.1722

This = the limit as n goes to infinity, 8 × 8 is 64, n × n³ is n⁴ × the summation of i goes from 1 to n of i³.1728

The index that I choose, that is the variable.1748

Everything else, I can pull out is a constant.1750

This is this, now I have something already, I know what this is.1753

The sum as i goes from 1 to n of i³, that is just equal to 1³ + 2³ + all the way to n³.1760

They gave me an expression for that, they said it is this.1772

That = n × n + 1/ 2².1776

I’m going to put this in for that and I get the following.1784

I get a = the limit as n goes to infinity of 64/ n⁴ × n × n + 1/ 2²1792

= the limit as n goes to infinity of 64/ n⁴ × n² × n² + 2n + 1/ 4,1811

= the limit as n goes to infinity of 64/ n⁴ × n⁴ + 2n³ + n²/ 4.1832

64 divided by 4, this is going to equal the limit as n goes to infinity of 16 × n⁴/ n⁴ is 1.1865

2n³/ n⁴ is 2/n.1879

N²/ n⁴ is n².1885

When I take n to infinity, this term goes to 0, this term goes to 0.1892

That is it, nice and simple.1902

I have my area = the limit as n goes to infinity of the sum i to n of my Δx × my f(x sub i).1904

I found an expression for Δx, I put it in.1917

I used my x sub 1, x sub 2, x sub 3, x sub 4, to elucidate some relationship,1921

in terms of n and I that I can put in for f(x sub i).1927

I put it in, I solved this, pulled out what I needed leaving just i.1932

As much as I can, i, underneath the summation symbol.1939

I had a closed form expression for that summation symbol.1942

I solved that, combined by simplifying.1945

And then, my final step, I just took the limit is n goes to infinity.1948

This goes to 0, this goes to 0, the limit is 16.1952

This is a long way of doing this.1958

In a couple of lessons we will find a very quick way of doing,1960

but it is really important that we understand where this comes from.1963

We did not just pull it out of nowhere.1967

Clearly, a lot of intricate detailed work goes into defining some of these higher mathematical concepts,1968

like finding area under a curve which we are going to call integration,1975

which is going to be essentially anti-differentiation.1979

Thank you so much for joining us here at www.educator.com.1983

We will see you next time, bye.1986

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