Professor Switkes

Area Between Curves

Slide Duration:

Table of Contents

Section 1: Overview of Functions

Review of Functions

26m 29s

Intro

0:00

What is a Function

0:10

Domain and Range

0:21

Vertical Line Test

0:31

Example: Vertical Line Test

0:47

Function Examples

1:57

Example: Squared

2:10

Example: Natural Log

2:41

Example: Exponential

3:21

Example: Not Function

3:54

Odd and Even Functions

4:39

Example: Even Function

5:10

Example: Odd Function

5:53

Odd and Even Examples

6:48

Odd Function

6:55

Even Function

8:43

Increasing and Decreasing Functions

10:15

Example: Increasing

10:42

Example: Decreasing

10:55

Increasing and Decreasing Examples

11:41

Example: Increasing

11:48

Example: Decreasing

12:33

Types of Functions

13:32

Polynomials

13:45

Powers

14:06

Trigonometric

14:34

Rational

14:50

Exponential

15:13

Logarithmic

15:29

Lecture Example 1

15:55

Lecture Example 2

17:51

Additional Example 3

-1

Additional Example 4

-2

Compositions of Functions

12m 29s

Intro

0:00

Compositions

0:09

Alternative Notation

0:32

Three Functions

0:47

Lecture Example 1

1:19

Lecture Example 2

3:25

Lecture Example 3

6:45

Additional Example 4

-1

Additional Example 5

-2

Section 2: Limits

Average and Instantaneous Rates of Change

20m 59s

Intro

0:00

Rates of Change

0:11

Average Rate of Change

0:21

Instantaneous Rate of Change

0:33

Slope of the Secant Line

0:46

Slope of the Tangent Line

1:00

Lecture Example 1

1:14

Lecture Example 2

6:36

Lecture Example 3

11:30

Additional Example 4

-1

Additional Example 5

-2

Limit Investigations

22m 37s

Intro

0:00

What is a Limit?

0:10

Lecture Example 1

0:56

Lecture Example 2

5:28

Lecture Example 3

9:27

Additional Example 4

-1

Additional Example 5

-2

Algebraic Evaluation of Limits

28m 19s

Intro

0:00

Evaluating Limits

0:09

Lecture Example 1

1:06

Lecture Example 2

5:16

Lecture Example 3

8:15

Lecture Example 4

12:58

Additional Example 5

-1

Additional Example 6

-2

Formal Definition of a Limit

23m 39s

Intro

0:00

Formal Definition

0:13

Template

0:55

Epsilon and Delta

1:24

Lecture Example 1

1:40

Lecture Example 2

9:20

Additional Example 3

-1

Additional Example 4

-2

Continuity and the Intermediate Value Theorem

19m 9s

Intro

0:00

Continuity

0:13

Continuous

0:16

Discontinuous

0:37

Intermediate Value Theorem

0:52

Example

1:22

Lecture Example 1

2:58

Lecture Example 2

9:02

Additional Example 3

-1

Additional Example 4

-2

Section 3: Derivatives, part 1

Limit Definition of the Derivative

22m 52s

Intro

0:00

Limit Definition of the Derivative

0:11

Three Versions

0:13

Lecture Example 1

1:02

Lecture Example 2

4:33

Lecture Example 3

6:49

Lecture Example 4

10:11

Additional Example 5

-1

Additional Example 6

-2

The Power Rule

26m 1s

Intro

0:00

Power Rule of Differentiation

0:14

Power Rule with Constant

0:41

Sum/Difference

1:15

Lecture Example 1

1:59

Lecture Example 2

6:48

Lecture Example 3

11:22

Additional Example 4

-1

Additional Example 5

-2

The Product Rule

14m 54s

Statement of the Product Rule

0:08

Lecture Example 1

0:41

Lecture Example 2

2:27

Lecture Example 3

5:03

Additional Example 4

-1

Additional Example 5

-2

The Quotient Rule

19m 17s

Intro

0:00

Statement of the Quotient Rule

0:07

Carrying out the Differentiation

0:23

Quotient Rule in Words

1:00

Lecture Example 1

1:19

Lecture Example 2

4:23

Lecture Example 3

8:00

Additional Example 4

-1

Additional Example 5

-2

Applications of Rates of Change

17m 43s

Intro

0:00

Rates of Change

0:11

Lecture Example 1

0:44

Lecture Example 2

5:16

Lecture Example 3

7:38

Additional Example 4

-1

Additional Example 5

-2

Trigonometric Derivatives

26m 58s

Intro

0:00

Six Basic Trigonometric Functions

0:11

Patterns

0:47

Lecture Example 1

1:18

Lecture Example 2

7:38

Lecture Example 3

12:15

Lecture Example 4

14:25

Additional Example 5

-1

Additional Example 6

-2

The Chain Rule

23m 47s

Intro

0:00

Statement of the Chain Rule

0:09

Chain Rule for Three Functions

0:27

Lecture Example 1

1:00

Lecture Example 2

4:34

Lecture Example 3

7:23

Additional Example 4

-1

Additional Example 5

-2

Inverse Trigonometric Functions

27m 5s

Intro

0:00

Six Basic Inverse Trigonometric Functions

0:10

Lecture Example 1

1:11

Lecture Example 2

8:53

Lecture Example 3

12:37

Additional Example 4

-1

Additional Example 5

-2

Equation of a Tangent Line

15m 52s

Intro

0:00

Point Slope Form

0:10

Lecture Example 1

0:47

Lecture Example 2

3:15

Lecture Example 3

6:10

Additional Example 4

-1

Additional Example 5

-2

Section 4: Derivatives, part 2

Implicit Differentiation

30m 5s

Intro

0:00

Purpose

0:09

Implicit Function

0:20

Lecture Example 1

0:32

Lecture Example 2

7:14

Lecture Example 3

11:22

Lecture Example 4

16:43

Additional Example 5

-1

Additional Example 6

-2

Higher Derivatives

13m 16s

Intro

0:00

Notation

0:08

First Type

0:19

Second Type

0:54

Lecture Example 1

1:41

Lecture Example 2

3:15

Lecture Example 3

4:57

Additional Example 4

-1

Additional Example 5

-2

Logarithmic and Exponential Function Derivatives

17m 42s

Intro

0:00

Essential Equations

0:12

Lecture Example 1

1:34

Lecture Example 2

2:48

Lecture Example 3

5:54

Additional Example 4

-1

Additional Example 5

-2

Hyperbolic Trigonometric Function Derivatives

14m 30s

Intro

0:00

Essential Equations

0:15

Six Basic Hyperbolic Trigc Functions

0:32

Six Basic Inverse Hyperbolic Trig Functions

1:21

Lecture Example 1

1:48

Lecture Example 2

3:45

Lecture Example 3

7:09

Additional Example 4

-1

Additional Example 5

-2

Related Rates

29m 5s

Intro

0:00

What Are Related Rates?

0:08

Lecture Example 1

0:35

Lecture Example 2

5:25

Lecture Example 3

11:54

Additional Example 4

-1

Additional Example 5

-2

Linear Approximation

23m 52s

Intro

0:00

Essential Equations

0:09

Linear Approximation (Tangent Line)

0:18

Example: Graph

1:18

Differential (df)

2:06

Delta F

5:10

Lecture Example 1

6:38

Lecture Example 2

11:53

Lecture Example 3

15:54

Additional Example 4

-1

Additional Example 5

-2

Section 5: Application of Derivatives

Absolute Minima and Maxima

18m 57s

Intro

0:00

Minimums and Maximums

0:09

Absolute Minima and Maxima (Extrema)

0:53

Critical Points

1:25

Lecture Example 1

2:58

Lecture Example 2

6:57

Lecture Example 3

10:02

Additional Example 4

-1

Additional Example 5

-2

Mean Value Theorem and Rolle's Theorem

20m

Intro

0:00

Theorems

0:09

Mean Value Theorem

0:13

Graphical Explanation

0:36

Rolle's Theorem

2:06

Graphical Explanation

2:28

Lecture Example 1

3:36

Lecture Example 2

6:33

Lecture Example 3

9:32

Additional Example 4

-1

Additional Example 5

-2

First Derivative Test, Second Derivative Test

27m 11s

Intro

0:00

Local Minimum and Local Maximum

0:14

Example

1:01

First and Second Derivative Test

1:26

First Derivative Test

1:36

Example

2:00

Second Derivative Test (Concavity)

2:58

Example: Concave Down

3:15

Example: Concave Up

3:54

Inconclusive

4:19

Lecture Example 1

5:23

Lecture Example 2

12:03

Lecture Example 3

15:54

Additional Example 4

-1

Additional Example 5

-2

L'Hopital's Rule

23m 9s

Intro

0:00

Using L'Hopital's Rule

0:09

Informal Definition

0:34

Lecture Example 1

1:27

Lecture Example 2

4:00

Lecture Example 3

5:40

Lecture Example 4

9:38

Additional Example 5

-1

Additional Example 6

-2

Curve Sketching

40m 16s

Intro

0:00

Collecting Information

0:15

Domain and Range

0:17

Intercepts

0:21

Symmetry Properties (Even/Odd/Periodic)

0:33

Asymptotes (Vertical/Horizontal/Slant)

0:45

Critical Points

1:15

Increasing/Decreasing Intervals

1:24

Inflection Points

1:38

Concave Up/Down

1:52

Maxima/Minima

2:03

Lecture Example 1

2:58

Lecture Example 2

10:52

Lecture Example 3

17:55

Additional Example 4

-1

Additional Example 5

-2

Applied Optimization

25m 37s

Intro

0:00

Real World Problems

0:08

Sketch

0:11

Interval

0:20

Rewrite in One Variable

0:26

Maximum or Minimum

0:34

Critical Points

0:42

Optimal Result

0:52

Lecture Example 1

1:05

Lecture Example 2

6:12

Lecture Example 3

13:31

Additional Example 4

-1

Additional Example 5

-2

Newton's Method

25m 13s

Intro

0:00

Approximating Using Newton's Method

0:10

Good Guesses for Convergence

0:32

Lecture Example 1

0:49

Lecture Example 2

4:21

Lecture Example 3

7:59

Additional Example 4

-1

Additional Example 5

-2

Section 6: Integrals

Approximating Areas and Distances

36m 50s

Intro

0:00

Three Approximations

0:12

Right Endpoint, Left Endpoint, Midpoint

0:22

Formulas

1:05

Velocity and Distance

1:35

Lecture Example 1

2:28

Lecture Example 2

12:10

Lecture Example 3

19:43

Additional Example 4

-1

Additional Example 5

-2

Riemann Sums, Definite Integrals, Fundamental Theorem of Calculus

22m 2s

Intro

0:00

Important Equations

0:22

Riemann Sum

0:28

Integral

1:58

Integrand

2:35

Limits of Integration (Upper Limit, Lower Limit)

2:43

Other Equations

3:05

Fundamental Theorem of Calculus

4:00

Lecture Example 1

5:04

Lecture Example 2

10:43

Lecture Example 3

13:52

Additional Example 4

-1

Additional Example 5

-2

Substitution Method for Integration

23m 19s

Intro

0:00

U-Substitution

0:13

Important Equations

0:30

Purpose

0:36

Lecture Example 1

1:30

Lecture Example 2

6:17

Lecture Example 3

9:00

Lecture Example 4

11:24

Additional Example 5

-1

Additional Example 6

-2

Section 7: Application of Integrals, part 1

Area Between Curves

19m 59s

Intro

0:00

Area Between Two Curves

0:12

Graphic Description

0:34

Lecture Example 1

1:44

Lecture Example 2

5:39

Lecture Example 3

8:45

Additional Example 4

-1

Additional Example 5

-2

Volume by Method of Disks and Washers

24m 22s

Intro

0:00

Important Equations

0:16

Equation 1: Rotation about x-axis (disks)

0:27

Equation 2: Two curves about x-axis (washers)

3:38

Equation 3: Rotation about y-axis

5:31

Lecture Example 1

6:05

Lecture Example 2

8:28

Lecture Example 3

11:55

Additional Example 4

-1

Additional Example 5

-2

Volume by Method of Cylindrical Shells

30m 29s

Intro

0:00

Important Equations

0:50

Equation 1: Rotation about y-axis

1:04

Equation 2: Rotation about y-axis (2 curves)

7:34

Equation 3: Rotation about x-axis

8:15

Lecture Example 1

8:57

Lecture Example 2

14:26

Lecture Example 3

18:15

Additional Example 4

-1

Additional Example 5

-2

Average Value of a Function

16m 31s

Intro

0:00

Important Equations

0:11

Origin of Formula

0:34

Lecture Example 1

2:51

Lecture Example 2

5:30

Lecture Example 3

8:13

Additional Example 4

-1

Additional Example 5

-2

Section 8: Extra

Graphs of f, f', f''

23m 58s

Intro

0:00

Slope Function of f(x)

0:41

Slope is Zero

0:53

Slope is Positive

1:03

Slope is Negative

1:13

Slope Function of f'(x)

1:31

Slope is Zero

1:42

Slope is Positive

1:48

Slope is Negative

1:54

Lecture Example 1

2:23

Lecture Example 2

8:06

Lecture Example 3

12:36

Additional Example 4

-1

Additional Example 5

-2

Slope Fields for Differential Equations

18m 32s

Intro

0:00

Things to Remember

0:13

Graphic Description

0:42

Lecture Example 1

1:44

Lecture Example 2

6:59

Lecture Example 3

9:46

Additional Example 4

-1

Additional Example 5

-2

Separable Differential Equations

17m 4s

Intro

0:00

Differential Equations

0:10

Focus on Exponential Growth/Decay

0:27

Separating Variables

0:47

Lecture Example 1

1:35

Lecture Example 2

6:41

Lecture Example 3

9:36

Additional Example 4

-1

Additional Example 5

-2

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For more information, please see full course syllabus of College Calculus: Level I

College Calculus: Level I Area Between Curves

Section 7: Application of Integrals, part 1: Lecture 1 | 19:59 min

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Lecture Comments (6)

	0 answers Post by Eti Sinha on September 8, 2014 For additional example #5, is it possible to find the area underneath the curve by: Integral of ( 1/y ) dy, from 1/3 to 3? Using dy as opposed to dx?
	0 answers Post by Rufino Hernandez on July 5, 2014 Asides from getting the area of this triangle using this method, you can also get it using another method. The Area of a triangle is square root of (s(s-a)(s-b)(s-c)). Where s=(1/2)(a+b+c). Where a is one side of the triangle, b is another side of the triangle, and c is the third side of the triangle. Since the Triangle's lower angle is the same as the angle made by the function y=\|x\|, it can be deduced that the angle made is 90 degrees. If it is a 90 degree triangle, then we can simply use the Area of a triangle to come to the conclusion of the area; A=(1/2)bh, where b=base and h=height. Since we have the end points of the sides of the triangle, we can simply derive sides. For example, the right side has an endpoint of (2,2); y=2, x=2. Using this we can make a second triangle where the sides are 2 and 2 which can lead us to find the third side which is square root of (8) = 2 square root of(2). We now know the length of the right side of the triangle. Using this same same method, we can derive the length of the base of the triangle (bottom left). The endpoint of this side is (-2/3, -2/3). So we can make a third triangle that has these sides and derive the length of the second side of the original triangle. This second side will be (-2/3)^2+(-2/3)^2=c^2 ==> (4/9)+(4/9)= c^2 ==> 8/9=c^2 ==> c= (2 * square root of (2))/3. Now since we have the height and base, we can simply put this into the Area of a triangle; A=(1/2)(base)(height) ==> A=(1/2)((2square root of (2))/2)(2 * square root of (2) ==> 4/3. Or we can solve for the third side of the triangle and input them into the (The Area of a triangle is square root of (s(s-a)(s-b)(s-c)). Where s=(1/2)(a+b+c). Where a is one side of the triangle, b is another side of the triangle, and c is the third side of the triangle.) equation and get the same results. This method will yield the same answer as the method stated in this video.
	0 answers Post by Fadel Hanoun on December 17, 2013 What about when x = 5pi/4, where Tan = 1?
	0 answers Post by Troy Ling on February 4, 2011 k= -15/16
	1 answer Last reply by: Nopparat SAntisathaporn Wed Oct 6, 2010 2:36 AM Post by Nopparat SAntisathaporn on October 6, 2010 Hi there, I don't know how to do number 7 in this test http://www.nzqa.govt.nz/nqfdocs/ncea-resource/exams/2007/90286-exm-07.pdf Can you please explain how to solve for K. Thank You.

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Area Between Curves

I recommend always starting with a sketch and drawing in a “sample rectangle.”
If an interval is not given, you may need to set the two functions equal in order to determine the interval involved.
If two curves cross, then you will need to break up the integral into more than one integral.
Sometimes, geometrical considerations can help you to check your results or even provide a non-calculus way to finish a problem.

Area Between Curves

Find the area enclosed by y = 4 and y = x²

First, we find the bounds. To do so, we set the equations equal to each other
x² = 4
x = ±2
A = ∫₋₂² 4 − x² dx
A = (4x − [(x³)/3]) |₋₂²
A = 8 − [8/3] − (−8 − [(−8)/3])
A = 16 − [16/3]

A = [32/3]

Find the area under the velocity curve v(t) = 3t² + t from t = 0 to t = 2

A = ∫₀² 3t² + t − 0 dt
The subtraction by zero has no effect. It is just to illustrate that we're taking the x-axis as the lower-bound.
A = (t³ + [(t²)/2]) |₀²
A = 8 + 2 − 0 − 0
The area under a velocity curve also happens to be the displacement!
∫_t₁^t₂ v(t) dt = x(t₂) − x(t₁)

A = 10

Find the area under y = e^x from x = 0 to x = ln10

A = ∫₀^ln10 e^x dx
A = e^x |₀^ln10
A = e^ln10 − e⁰
A = 10 − 1
A = 9

A = 9

Find the area under y = e^[t/3] from t = 0 to t = 6

A = ∫₀⁶ e^[t/3] dt
u = [t/3]
du = [1/3] dt
A = 3 ∫₀⁶ e^u du
A = 3 e^[t/3] |₀⁶
A = 3 (e² − e⁰)

A = 3(e² − 1)

Find the area bounded by y = x² and y = |x|

Find the bounds of the integral
x² = |x|
x² = √{x²}
x⁴ = x²
x² = 1
x = ±1
y = |x| has different behavior on either side of zero. We can treat this as a piecewise function with the area as the sum of two different integrals
A = ∫₋₁⁰ −x − x² dx + ∫₀¹ x − x² dx
A = (−[(x²)/2] − [(x³)/3]) |₋₁⁰ + ([(x²)/2] − [(x³)/3]) |₀¹
A = (−0 − 0) − (−[((−1)²)/2] − [((−1)³)/3]) + ([1/2] − [1/3]) − 0
A = [1/2] − [1/3] + [1/2] − [1/3]
A = 1 − [2/3]

A = [1/3]

Find the area bound by x = y³ and the y-axis from y = 0 to y = 2

A = ∫₀² y³ dy
A = [(y⁴)/4] |₀²
A = [(2⁴)/4] − 0

A = 4

Find the area bound by y = sinx and the x-axis from x = 0 to x = 2π

A = ∫₀^2π sinx dx
A = − cosx |₀^2π
A = −cos2π− (− cos0)
A = −1 + 1 = 0
An area of zero doesn't quite make sense here, so using the straight integral is insufficient. What we can do is treat this as two separate integrals, one where the area is above the x-axis and one where it is below and add their effective area.
sinx = 0, x = 0, π, 2π in this interval.
A = ∫₀^π sinx dx + (−∫_π^2π sinx dx)
The negative sign is there to compensate for the "negative area"
A = −cosx |₀^π + cosx |_π^2π
A = − cosπ+ cos0 + cos2π− cosπ
A = 1 + 1 + 1 + 1

A = 4

Find the area bound by x = y² and y = x − 2

We could find the integral in terms of dx, but the we would have to split it up into at least two separate integrals. Besides, x = y² is not a function of x (there are two outputs for every x input).
x = y²
x = y + 2
Find the limits of the integral
y² = y + 2
y² − y − 2 = 0
(y + 1)(y − 2) = 0
y = 2, −1
A = ∫₋₁² (y + 2) − y² dy
A = ([(y²)/2] + 2x − [(y³)/3]) |₋₁²
A = (2 + 4 − [8/3]) − ([1/2] − 2 + [1/3])
A = 8 − [1/2] − [9/3]
The area above the

A = [9/2]

Find the area bound by y = x² + 1 and y = −x² + 3

Find the limits of the integral
x² + 1 = −x² + 3
2x² = 2
x² = 1
x = ±1
A = ∫₋₁¹ (−x² + 3) − (x² + 1) dx
A = ∫₋₁¹ −2x² + 2 dx
A = (−[(2x³)/3] + 2x) |₋₁¹
A = −[2/3] + 2 − (−[(−2)/3] − 2)
A = −[2/3] + 2 − [2/3] + 2
A = 4 − [4/3]

A = [8/3]

Find the area bound by y = x² and y = 2x

Find the limits
x² = 2x
x² − 2x = 0
x(x − 2) = 0
x = 0, 2
A = ∫₀² 2x − x² dx
A = x² − [(x³)/3] |₀²
A = 4 − [8/3] − 0

A = [4/3]

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Answer

Area 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

Area Between Two Curves

Graphic Description

Lecture Example 1

Lecture Example 2

Lecture Example 3

Additional Example 4

Additional Example 5

Intro 0:00
Area Between Two Curves 0:12

Graphic Description

Lecture Example 1 1:44
Lecture Example 2 5:39
Lecture Example 3 8:45
Additional Example 4
Additional Example 5

College Calculus 1 Online Course

Section 1: Overview of Functions
	Review of Functions	26:29
	Compositions of Functions	12:29
Section 2: Limits
	Average and Instantaneous Rates of Change	20:59
	Limit Investigations	22:37
	Algebraic Evaluation of Limits	28:19
	Formal Definition of a Limit	23:39
	Continuity and the Intermediate Value Theorem	19:09
Section 3: Derivatives, part 1
	Limit Definition of the Derivative	22:52
	The Power Rule	26:01
	The Product Rule	14:54
	The Quotient Rule	19:17
	Applications of Rates of Change	17:43
	Trigonometric Derivatives	26:58
	The Chain Rule	23:47
	Inverse Trigonometric Functions	27:05
	Equation of a Tangent Line	15:52
Section 4: Derivatives, part 2
	Implicit Differentiation	30:05
	Higher Derivatives	13:16
	Logarithmic and Exponential Function Derivatives	17:42
	Hyperbolic Trigonometric Function Derivatives	14:30
	Related Rates	29:05
	Linear Approximation	23:52
Section 5: Application of Derivatives
	Absolute Minima and Maxima	18:57
	Mean Value Theorem and Rolle's Theorem	20:00
	First Derivative Test, Second Derivative Test	27:11
	L'Hopital's Rule	23:09
	Curve Sketching	40:16
	Applied Optimization	25:37
	Newton's Method	25:13
Section 6: Integrals
	Approximating Areas and Distances	36:50
	Riemann Sums, Definite Integrals, Fundamental Theorem of Calculus	22:02
	Substitution Method for Integration	23:19
Section 7: Application of Integrals, part 1
	Area Between Curves	19:59
	Volume by Method of Disks and Washers	24:22
	Volume by Method of Cylindrical Shells	30:29
	Average Value of a Function	16:31
Section 8: Extra
	Graphs of f, f', f''	23:58
	Slope Fields for Differential Equations	18:32
	Separable Differential Equations	17:04

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	Calculus by Larson, 9th Edition Authors: Ron Larson, Bruce H. Edwards ISBN: 0547167024 Publisher: Brooks Cole Year: 2009 The book is filled with examples and detailed explanations. Capstone exercises are also included for each section and synthesize the main concepts into a single example.
	Calculus by Larson, 9th Edition, Solutions Manual, Volume 1 Author: Ron Larson ISBN: 547213093 Publisher: Brooks Cole Year: 2008 This solutions manual includes worked out solutions to every odd-numbered exercise in Calculus of a Single Variable, 9th edition.
	Calculus by Larson, 9th Edition, Solutions Manual, Volume 2 Authors: Ron Larson, Bruce H. Edwards ISBN: 547213107 Publisher: Brooks Cole Year: 2009 This solutions manual includes worked out solutions to every odd-numbered exercise in Multivariable Calculus, 9th edition.

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College Calculus: Level I Area Between Curves

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College Calculus: Level I Area Between Curves

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