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INSTRUCTORS Raffi Hovasapian John Zhu
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Raffi Hovasapian

Raffi Hovasapian

Example Problems for Areas Between Curves

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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Lecture Comments (6)

1 answer

Last reply by: Professor Hovasapian
Wed May 11, 2016 2:55 AM

Post by Acme Wang on April 27, 2016

In Example II, why the antiderivative of e^(1/2y)equals 2e^(1/2y)?

PS: I am a super fan of your calculus class! It does help me a lot!!! Thank you!

1 answer

Last reply by: Professor Hovasapian
Sat Apr 23, 2016 6:45 PM

Post by Cam-Tuoi Dinh on April 22, 2016

In example 8, why the unit of the answer for problem d has to be in second, not in minutes?

1 answer

Last reply by: Professor Hovasapian
Sat Mar 26, 2016 4:49 AM

Post by Sazzadur Khan on March 6, 2016

On example 3, shouldn't the evaluated integral read 20x-x^3?

Example Problems for Areas Between Curves

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
  • 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?
    • Part B: If We Shaded the Region between the Graphs from t=0 to t=2.187, What Would This Shaded Area Represent?
    • Part C: At 4 Minutes Which Car is Ahead?
    • Part D: At What Time Will the Cars be Side by Side?

Transcription: Example Problems for Areas Between Curves

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

Today, we are going to be doing some example problems for areas between curves.0004

Let us jump right on in.0008

In the following examples, we want you to sketch the given curves, identify the region that is enclosed by these curves.0011

State whether you are going to be integrating with respect to x or y.0020

Draw a representative rectangle, if you can.0025

Finally, find the area of this region, or at the very least set up the integral for the area of this region.0028

Let us get started.0035

Our first one is going to be y = 7x - x² and y = x.0039

Let us go ahead and draw this out and see what is it we are actually dealing with.0043

I think I will go ahead and work in blue.0048

Let me draw it over here.0054

How we are going to draw 7x - x²?0060

I'm going to go ahead and work over here.0064

I’m going to factor this, y = x × 7 - x and that is equal to 0, that gives me x = 0 and that gives me x = 7.0068

I know that the graph is going to hit a 0 and I know the graph is going to hit at 7.0083

The midpoint between 0 and 7, the midpoint which is where the vertex is going to be,0089

I know this is a -x², I know it is a thing that is going to open down.0101

Therefore, I know it is going to be going up this way and up this way.0105

The midpoint = 3.5.0111

When I put 3.5 into the original, I end up with y = 12.25.0114

At 3.5, 12.25, that is that graph right there.0123

As far as the y = x is concerned, we know that one, that is just a line that way.0134

The region that we are interested in is this region right here.0141

In this particular case, I’m going to go ahead and draw a representative rectangle.0149

The rectangle is vertical, I'm going to be adding the rectangles this way.0155

I'm going to be integrating along x.0159

What is the x value of this point?0172

We are going to be integrating from 0 all the way to the x value of this point, whatever that is.0176

We need to find that.0186

What is the x value of this point?0187

I’m going to set them equal to each other.0197

I have got 7x - x² is equal to x.0198

I have got x² - 6x = 0.0208

I have got x × x - 6 = 0 which gives me x = 0.0216

That is one of them.0221

x = 6, that is the other point.0224

Remember, this one was 7.0228

We integrate from 0 to 6, there we go.0234

From here to here, we are going to add up all of the individual rectangles.0244

That is going to give us the area of the curve.0250

Let us go ahead and do that.0252

With each of these, I actually rendered the picture itself.0256

We have a nice drawing to look at.0260

This is the region that we are interested in, in between here and here.0263

This is our 0, this is our 6, therefore, the area is going to be the integral from 0 to 6, upper function - the lower function.0268

The upper function is 7x - x² - the lower function which is x.0281

I’m going to integrate along x so it is dx.0290

This is the integral from 0 to 6 of 7x – x.0294

I can simplify 6x - x² dx.0299

This is going to equal 6x²/ 2 – x³/ 3.0306

I'm going to evaluate this from 0 to 6.0316

My answer is going to be 36, that is it.0319

I put 6 into here.0324

I put 6 into here and I end up with 108 – 72, that is for the 6.0342

And then, -, 0 – 0 is for the 0, that gives me my 36.0350

Again, you can go ahead and actually evaluate it.0358

You can go ahead and use your calculator.0361

Remember, we looked at that earlier.0363

Just enter the function, do second calc.0366

Go down to integration, lower limit, upper limit.0371

You have got yourself your integral value.0375

In this particular case, it is really easy to evaluate without the calculator, but in case you want to do that.0378

You got 36.0382

Example number 2, we have x = y² – 3, x = e¹/2 y.0388

We have y = -1 and y = 2.0394

In this particular case, it is y that is the independent variable.0397

We are going to be looking at functions which are left to right, instead of up down.0400

Let us take a look at what it is that we have got going here.0404

x = y² – 3.0410

My y² graph looks like this, the -3 part means it is a shift to the left because it is a function of y.0412

It is with left, my graph is actually going to be 1, 2, 3.0423

It is going to look something like that.0430

x = e ^½ y, when y is equal to 0, e⁰ is 1, that means x is 1.0435

It is going to be one of my points.0452

Let me go ahead and mark 1, 2.0455

Let us go 1, 2, let us go 3.0476

When y = 2 it is 2/2, it is 1 e ⁺12.718.0486

When y = 2, I got to x 0.718, it is going to be somewhere around there.0496

Basically, what I’m going to get is this, it is not going to cross, it is asymptotic right there.0508

It is going to look something like that.0518

y = -1 that this line, y = 2 is this line.0521

The region that I'm interested in is going to be this region right here.0532

I have a better picture on the next page.0539

It is going to be something like that.0542

We are going to be integrating from -1 to 2.0545

We are going to be integrating along the y axis.0549

Our representative rectangle is going to look something like that.0553

I’m going to be integrating vertically.0557

We will say best to integrate along the y axis from -1 to 2.0564

Therefore, the area is going to be the integral from -1 to 2, right function - left function.0586

We are integrating along y.0606

Let us take a look at a better picture here.0610

Yes, this is the region we are interested in.0612

That, that, that, there.0615

We want this and we are integrating horizontally from -1 to 2.0624

Therefore, our area is going to equal the integral from -1 to 2.0636

The right function is going to be e ^½ y - the left function y² - 3 dy.0647

That is going to equal the integral from -1 to 2 of e ^½ y - y² + 3 dy.0664

That is going to equal 2 e ^½ y - y³/ 3 + 3y evaluated from -1 to 2.0680

I went ahead and I just did this via my calculator.0702

However, if you want to see what it actually looks like, when you plug 2 in here, you are going to get 3 terms.0704

-1 in here, you are going to get 3 terms.0709

I got 2e – 8/3 + 6, that is when I plug the 2 in, and then I subtract.0712

When I plug in the -1, it is going to be 2e⁻¹/2 + 1/3 – 3.0722

If I have done everything correctly, and my final answer that I got was 10.224.0733

That is it, nice and straightforward.0740

Right function - left function, in this case.0743

y = 1/x, y = 1/x³, x = 4.0747

This one should be reasonably straightforward, let us do this0753

Let us go 1, 1, 1/x.0760

Let me make this a little bit bigger.0767

I’m going to put the 1,1 right there.0773

I have got something like that, that is my y = 1/x, my 1/x³.0775

Something like that, that was my y = 1/x³.0789

If I go to 4, something like that.0795

It looks like I’m going to be integrating from 1 to 4.0800

I'm going to be integrating, this is one of my representative rectangles.0806

I'm going to be adding the rectangles horizontally, I’m integrating along the x axis.0809

We integrate along x from 1 to 4.0818

Let us take a look at a picture here.0825

Nice, better, looking picture.0828

This is the region, here is our 1, here is our 4.0830

Here is our representative rectangle and we are adding this way.0837

What I have got is the area = the integral from 1 of 4 of the upper function - the lower function.0842

The upper function is this one, that is the 1/x - 1/x³ dx.0861

It is going to be integral from 1 to 4 of 1/x dx - the integral from 1 to 4 of x⁻³ dx.0863

It is going to equal, this one is going to be ln(x) evaluated from 1 to 4, -x⁻²/ -2 evaluated from 1 to 4.0876

If I have done everything correctly, I get ln 4 - ln 1 - -1/32 - -1/2, this goes to 0.0896

When I add everything up, I get 0.9175.0917

Again, it all comes down to the same thing.0924

It is going to be the upper function – the lower function or right function - left function.0926

All you have to do is find the limits of integration.0932

The limits of integration are going to come from either ends that are given to you, in this case x = 4.0935

The x value or the y value for where the two graphs happens to meet.0945

That is the only thing that is going on.0949

15 – 2x², y = x² – 5, I mentioned the biggest difficulty here is actually drawing these functions out,0955

if you remember them from pre-calculus.0962

Let us go ahead and draw this out.0965

15 - 2x² - 2x² opens down, let us see where it actually hits the x axis.0971

We have got 15 - 2x² is equal to 0.0979

You have got 2x² = 15, x² = 15/2, that means x is going to be + or -2.74.0986

It is going to be, I have got a point here, a point here.0999

I have one parabola that looks like this.1010

I have my other parabola which is going to be the x² – 5.1014

x² - 5 = 0, x² = 5, x = + or -2.24.1019

2.24 is like there and there, down to 5.1030

This parabola goes this way, this way.1035

I need the area in between those two curves.1042

It is going to be upper – lower.1048

Here is going to be one of my representative rectangles.1051

I'm going to integrate along the x axis.1053

My limits of integration, I need to know the x value of that point and the x value of that point, where those two meet.1056

Let us find that out, I find that out by setting the two functions equal to each other.1061

15 - 2x² = x² – 5.1068

I get 20 = 3x², x² = 20/3, x = + or -2.582.1074

This is 2.582, this is -2.582.1089

Those are going to be my limits of integration.1093

I'm going to take the integral, the area is going to be integral from - 2.582 to +2.582.1096

The upper function - the lower function dx.1106

Let us see what that looks like.1115

There you go, here is your -2.582, here is your +2.582.1118

I have got my area is equal to the integral -2.582.1133

You are definitely going to need to use a calculator for this.1140

2.582 of the upper function which is 15 - 2x² - the lower function which is x² - 5 dx,1143

which is equal to the integral -2.582 to +2.582.1158

It looks like I have got 20 - 3x² dx.1165

This is going to equal 20x - x² evaluated from that to that.1186

When I put the values in, I get 68.853.1201

There you go, that is the area between the curves.1209

Nice, simple, straightforward, not a problem.1218

x = 1/8 y³, x = 6 - y², let us go ahead and draw this out.1226

Again, it is probably best to just go ahead and use your calculator or something like www.desmos.com.1239

What you end up getting is, 6 - y² this is going to be a graph that looks like this.1246

x = ½ y³ is going to look something like this, something like that.1254

We are looking at this region right there.1263

It looks like it is best to go horizontal rectangles, which means we are going to integrate from bottom to top.1269

We want to integrate along y, we need the y value of that point and we need the y value of that point.1281

That is going to be our lower limit to our upper limit of integration.1292

Let us go ahead and do that, find that first.1296

We set the two functions equal to each other.1298

1/8 y³ = 6 - y².1302

I get y³ + 8y² - 48 is equal to 0.1309

We use our calculators, we use our software, we use Newton’s method,1319

whatever it is that we need to do to find the values of y to satisfy that.1323

We end up with two values, y = 2.172 and y = -3.144.1327

This value is our 2.172, that is our upper limit of integration.1343

This value is our -3.144, that is our lower limit of integration.1348

Our area is going to equal -3.144 to 2.172, right function - left function.1356

We are integrating along y, this is going to be, dy.1374

Here is the region that we are discussing.1381

In between there, that is our y value of 2.172.1390

Here is our y value of -3.144.1400

What have we got?1407

Let me go ahead and write it down here.1408

We have the area is equal to the integral -3.144 to 2.172, right function - left function.1412

I have got 6 - y² – 1/8 y³ dy.1424

I get 6y - y³/ 3 – y⁴/ 32.1436

I evaluate this from -3.144 to 2.172.1452

When I put it into a calculator, I get 20.479.1459

Nice and straightforward.1468

Let us do something that involves some sine and cosine.1475

We have y = cos x, y = sin 2x.1478

I would like you to integrate this or find the area the region between those two curves, between 0 and π/2.1481

Let us go ahead and draw this out.1490

Let us stay in the first quadrant here.1494

That is okay, I do not need to make it quite so big.1497

y = cos x, period is 2π, it is going to look something like this.1503

This is π/2.1512

y = sin(2x), the period is π.1517

If the period is π, that means it is going to start at 0 and1521

it is going to hit 0 again at π/2 that means at π/4, it is going to hit a high point.1526

Let us go ahead and call this 1.1532

We are going to get something like that, the region that we are interested in, this region right here.1535

What is the area of that region?1550

Notice, here this function, this is our cos x.1554

This function is our sin 2x, from 0 to some value which I will call a for now, cos x is above sin x, that is upper – lower.1561

Pass this point, it is the sin 2x that is actually the upper function and the cos x is the lower function.1573

I'm going to have to break this up into two integrals.1581

Integrate from 0 to a, cos x - sin 2x, from a to π/2, sin 2x - cos x.1583

We need to find what a is first.1592

How do you find what a is equal to?1596

You set the two functions equal to each other and you solve for x.1598

I have got cos(x) = sin(x), cos(x) -, sin(2x) there is an identity.1602

Sin(2x) = 2 sin x cos - 2 sin x cos x = 0.1615

I have moved it to the left and used my identity.1624

I’m going to factor out a cos x.1626

Cos x, 1 – 2, sin x = 0 that gives me two equations.1629

Cos x = 0 and sin x = ½.1637

Cos x = 0 that is going to be my π/2.1643

I already know that they meet there.1647

This value right here, sin(x) = ½.1649

Between 0 and π/2, my x = π/ 6, that is equal to my a.1653

My a is equal to π/6.1660

Therefore, I’m going to be integrating from 0 to a, 0 to π/6.1664

I’m going to be integrating from π/6 to π/2.1674

This first integral, second integral.1681

Let us go ahead and see what this looks like.1684

The region that I’m interested in is, a that is going to be my first integral.1688

This is going to be my second integral.1699

The red one is my cos(x), the blue one is my sin(2x).1703

Therefore, what I have got is area is equal to the integral from 0 to π/6 cos x - sin(2x) dx1711

+ the integral from π/6 to π/2.1726

Now it is sin 2x - cos x.1732

Nice and straightforward, the rest is just integration.1741

This is going to be sin x + ½ cos 2x evaluated from 0 to π/6 +1743

this is going to be -1/2 cos(2x) - sin x evaluated from π/6 to π/2.1761

When you actually evaluate this which I would not do here, you are going to get ½.1776

If I have done all of my arithmetic correctly, or if my calculator do the arithmetic correctly.1781

y = 2x² 10 x² 7x + 2y = 10.1795

You know what, this example, I think I'm actually just going to go through really quickly,1800

just run through it because I think we get the idea now.1805

Do not want to spend too much time hammering the point.1809

This is what the graph looks like.1812

Here is your graph of the y = 10x².1815

This is the 2x².1822

This time right here is the 7x + 2y = 10 which I have actually written as y = -7/2 x + 5.1825

You are going to solve it for y = mx + b, in order to graph it.1836

The region we are interested in is this region right here.1840

I’m going to go ahead and do this by breaking this up.1849

I’m going to do this is in two integrals.1853

I’m going to integrate along x.1854

I’m going to take the representative rectangle there for my first region and my second region.1857

I’m going to go from 0 to whatever this number is, which I will find in just a minute.1864

I'm going to go from this number to whatever this number is, for my second integral.1870

When I set y = 10x² equal to y = -7/2 x + 5, I’m going to end up this value right here.1879

My x value is going to be 0.553.1893

This point, I'm going to set my 2x² equal to my -7/2 x + 5.1898

This value I'm going to get is going to be 0.932.1905

Once I have that, the integral is really simple.1917

It is just going to be the area is equal to the integral from 0 to 0.553, upper function - lower function,1920

10x² - 2x² dx + 0.553 to 0.932 of the upper function which is -7/2 x + 5.1934

Because we are integrating with respect to x, it has to be a function of x - the lower function.1955

From here to here, that is the lower function which is the 2x² dx.1963

When I evaluate this integral and solve, I get 0.9341 as my area, that is it.1968

This is region 1, this is region 2, I just broke it up.1980

I have to find the x value for there, the x value for there, integrate from 0 to that point first1983

and that point first, upper – lower, upper – lower.1990

That is it, just do what is exactly what you think you should do.1993

It is very intuitive.2000

Let us go ahead and talk about this problem now.2006

Cars A and B start from rest and they accelerate.2008

The graphs below show their respective velocity vs. time graph × along the x axis,2012

the velocity of the cars is along the y axis.2018

Car A is the blue graph, this is A.2022

The graph, the value of the function is, this function right here is √2x.2025

B is the red graph, this is car B.2035

Its function, when expressed as a function of x is 1/5 x³.2039

First question we are going to ask, at 2.187 minutes where the graphs meet,2047

this point right here, this is our 2.187, which car is further ahead?2052

This is a velocity vs. time graph, when you integrate a velocity vs. time graph,2061

in other words, when you find the area under the curve up to a certain time, that gives you the total distance traveled.2067

Because again, velocity is, let us say meters per second, the differential time element is dt, meters per second.2075

The integral of v dt, velocity is expressed in, let us just say it is meters per second.2088

Time is expressed in second.2096

When you multiply those and add them all up, in other words integrate from 0 to 2.187,2099

you are going to get the area under the curve for car A.2105

You are going to get the area under the curve for car B, whichever area is bigger, that has gone further.2108

Clearly, car A has the bigger area, the area under all of the blue graph is a lot more than the area under the red graph.2117

Therefore, part A is really simple, it is car A is further ahead.2127

Car A is further ahead and this is further ahead because2134

the area under its graph is a lot more than area of the graph for car B which is just that right there.2140

If we shaded the region between the graphs from t = 0 to t = 187, what would the shaded area represent?2155

Let me go to black.2162

If I shaded in the area between the graphs, in other words this area, if I shaded that area, what does that represent?2164

The area on the blue graph is the total distance the blue has gone.2175

The area under the red graph is the total distance the red car has gone.2179

The difference between them is just how much further car A is then car B.2185

Part B, I will just write it up here.2193

Part B is what does it represent?2198

The shaded region represents the distance A is farther from the farther from B.2204

That is all, area under the graph for A, area under the graph for B,2218

the difference between them is the area between the two graphs.2225

It is how much father A is than B.2228

Part C, at 4 minutes, which car is ahead, and D, at what time will the cars be side by side?2233

When we look at this graph, we see that A, it accelerates faster and steadies out.2241

B, accelerates slower, it is further behind but at some point it really starts to accelerate.2248

Eventually, it is going to catch up.2255

A is going to be ahead of be but at some point B, is going to pass A.2256

At 4 minutes, which car is ahead?2261

The question we are asking is at 4 minutes which has a greater area under its graph? 4.2264

Let us go ahead and work that one out on the next page with a graph.2272

We are going to integrate, we are going to find the area under the blue graph from 0 to 4,2277

and we are going to find the area under the red graph from 0 to 4.2283

We said that the blue graph was √2x and we said this one was 1/5 x³.2288

From car A which is the blue graph, we have the area is equal to the integral from 0 to 4 of √2x dx.2299

When I solve that, I get 7.543.2319

For car B, the area = the integral from 0 to 4 of 1/5 x³ dx.2324

When I solve that, I get 12.8.2336

Car B, at 4 minutes or 4 seconds, whatever the time unit is, now car B has actually past car A.2340

Car B is farther forward.2347

The last question asked, at what time are they side by side?2355

Side by side means it is going the same distance.2361

At what time are the two areas equal?2364

We need the area of car A to equal the area of car B.2368

The area of car A is going to be the integral from 0 to l.2379

L is the time that we are looking for of √2x dx is equal 0 to l, that is the time period.2384

From 0 to whatever time we are looking for, we are looking for l, 1/5 x³ dx.2395

This is going to be √2 × the integral from 0 to l of x ^½ dx.2407

It is going to equal 1/5, the integral from 0 to l of x³ dx.2415

This is going to be, I’m going to come up here, 2√2/ 3 x³/2 from 0 to l is going to equal x⁴/ 20 from 0 to l.2426

When I put l in for here, when I put l in for here, 0 on for here, 0 on for here, set them equal to each other and solve.2451

I get something like this, I get 2√2/ 3 l³/2 = l⁴/20.2457

I need to solve for l.2472

The equation that this gives me is l⁴/20, let us bring this over this side, set it equal to 0 – 2√2/ 3 l³/2 = 0.2475

When I use my mathematical software or my calculator, or whatever it is to find what l is, I get l = 3.24 seconds.2498

I hope that made sense.2512

I want to know, when are they going to be side by side?2513

Side by side means they have gone the same distance.2517

Distance is the area underneath their respective graphs.2520

I set the areas equal to each other, the areas are the integrals.2523

L is my time, the upper limit integration.2528

I’m solving for l, I get an equation in l.2531

I solve for l, I get 3.24 seconds.2534

I did this by using a particular graph.2537

Here is what the graph looked like.2540

That function that we ended up getting, that this function is our l⁴/20 – 2√2/ 3 l³/2 = 0.2542

I just want to know where it is equal 0, 3.24 seconds.2556

You can use a calculator, you can use a graph, find where it crosses the x axis, anything you want.2560

I hope that make sense, and that is areas between curves.2567

Thank you so much for joining us here at www.educator.com.2572

We will see you next time, bye.2574

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