Professor Switkes

Volume by Method of Cylindrical Shells

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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College Calculus: Level I Volume by Method of Cylindrical Shells

Section 7: Application of Integrals, part 1: Lecture 3 | 30:29 min

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Lecture Comments (3)

0 answers

Post by Meghan Brunet on April 25, 2013

"=...=" is mind blowing and not very helpful.

0 answers

Post by Joanna Nicholson on February 16, 2013

This is a very good question, I asked it myself. I wish someone would answer the questions....

0 answers

Post by Jacqueline Measer on July 11, 2012

Why do you draw two different cylindrical shells? In class only one was drawn.

Answer EngineGet answers to any question!Ask any question related to College Calculus: Level I

Working on the solution...

Volume by Method of Cylindrical Shells

I highly recommend starting with a sketch of the functions involved and drawing in a sample radius!
I recommend memorizing the formulas but also understanding the geometry that leads to the formulas.
If an interval is not given, you may need to set the two functions equal in order to determine the interval involved.
Sometimes, geometrical considerations can help you to check your results or even provide a non-calculus way to finish a problem.

Volume by Method of Cylindrical Shells

Find the volume of the solid that results from revolving the area bounded by y = x², y = 0, and x = 4 around the y-axis.

Find the lower bound of the integral (we're already handed the upper bound for x)
x² = 0
x = 0
V = 2 π∫₀⁴ x (x²) dx
V = 2π[(x⁴)/4] |₀⁴
V = 2π[(4⁴)/4]

V = 128π

Find the volume of the solid that results from revolving the area bound by y = [1/x], x = 1, and x = 2 around the y-axis.

V = 2π∫₁² x[[1/x] − 0] dx
V = 2π∫₁² 1 dx
V = 2πx |₁²
V = 2π(2 − 1)

V = 2π

Find the volume of the solid that results from revolving the area bound by y = sinx², y = 0, and x = 1 around the y-axis.

x² = 0
x = 0
V = 2 π∫₀¹ x[sinx²] dx
V = 2 π∫₀¹ x sinx² dx
u = x²
du = 2x dx
V = 2 π[1/2] ∫₀¹ sinu du
V = π(−cosx²) |₀¹
V = π(−cos1 + cos0)
V = π(1 − cos1)

V = π(1 − cos1)

Find the volume of the solid that results from revolving the area bound by y = [1/2] e^x², y = 0, x = 0, and x = 1 around the y-axis.

V = 2 π∫₀¹ x [1/2] e^x² dx
u = x²
du = 2x dx
V = 2π[1/2] [1/2] ∫₀¹ e^u du
V = [(π)/2] e^u |₀¹
V = [(π)/2] e^x² |₀¹
V = [(π)/2] (e¹ − e⁰)

V = [(π)/2] (e¹ − 1)

Find the volume of the solid that results from revolving the area bound by y = √{1 − x²} and y = 0 from x = 0 to x = 1 around the y-axis.

The function is undefined for x <−1 and x > 1
Let's find the points where the two curves intersect
x-axis is y = 0
√{1 − x²} = 0
1 − x² = 0
x² = 1
x = −1, 1
The intersections and all values of x in between are defined
V = (2π) ∫₀¹ x √{1 − x²} dx
u = 1 − x²
du = −2x dx
V = 2π[(−1)/2] ∫₀¹ u^[1/2] du
V = −π[(u^[3/2])/([3/2])] |₀¹
V = [(−2π)/3] (1 − x²)^[3/2] |₀¹
V = [(−2π)/3] ( (1 − 1)^[3/2] − 1^[3/2])
We've found the volume of a half-sphere of radius 1

V = [(2 π)/3]

Find the volume of the solid that results from revolving the area bound by y = x³, y = 1, and x = 2 around the y-axis.

Find the x value where the two curves intersect for the other bound of the integral
x³ = 1
x = 1
V = 2π∫₁² x[x³ − 1] dx
V = 2π∫₁² x⁴ − x dx
V = 2π([(x⁵)/5] − [(x²)/2]) |₁²
V = 2π([32/5] − 2 − ([1/5] − [1/2]))

V = [(47π)/5]

Find the volume of the solid that results from revolving the area in between by y = [10 lnx/(x²)] and y = 0 from x = 1 to x = 6 around the y-axis.

V = 2 π∫₁⁶ x [[10 lnx/(x²)]] dx
V = 20 π∫₁⁶ [lnx/x] dx
u = lnx
du = [1/x] dx
V = 20 π∫₁⁶ u du
V = 20π[(u²)/2] |₁⁶
V = 20π[((lnx)²)/2] |₁⁶
V = 10 π((ln6)² − (ln1)²)

V = 10 π(ln6)²

Find the volume of the solid that results from revolving the area bound by y = [1/(√{4 − x²})], x = 0, x = 1 and y = 0 around the y-axis.

V = 2 π∫₀¹ x[[1/(√{4 − x²})]] dx
u = 4 − x²
du = −2x dx
V = 2 π[(−1)/2] ∫₀¹ [1/(√u)] du
V = −π∫₀¹ u^−[1/2] du
V = −π[1/([1/2])] u^[1/2] |₀¹
V = −2 π(4 − x²)^[1/2] |₀¹
V = −2π(√3 − √4)

V = 2π(2 − √3)

Find the volume of the solid that results from revolving the area bound by y = 2x, y = 0, and x = 4 around the x-axis.

Let's rewrite this in terms of x, since we're rotating about the x-axis
y = 2x
x = [1/2] y
Find the bounds of the integral
[1/2] y = 4
y = 8
V = 2 π∫₀⁸ y[4 − [1/2] y] dy
V = 2 π∫₀⁸ 4y − [1/2] y² dy
V = 2 π(2y² − [(y³)/6]) |₀⁸
V = 2 π(2(64) − [(8³)/6] − 0)
V = 2 π(128 − [512/6])

V = [(256π)/3]

Find the volume of the solid that results from revolving the area bound by x = y³ and x = y² around the x-axis.

y³ = y²
y = 0, 1
V = 2 π∫₀¹ y[y² − y³] dy
V = 2 π∫₀¹ y³ − y⁴ dy
V = 2 π([(y⁴)/4] − [(y⁵)/5]) |₀¹
V = 2 π([1/4] − [1/5] − 0)

V = [(π)/10]

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Answer

Volume by Method of Cylindrical Shells

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

Important Equations

Lecture Example 1

Lecture Example 2

Lecture Example 3

Additional Example 4

Additional Example 5

Intro 0:00
Important Equations 0:50

Equation 1: Rotation about y-axis
Equation 2: Rotation about y-axis (2 curves)
Equation 3: Rotation about x-axis

Lecture Example 1 8:57
Lecture Example 2 14:26
Lecture Example 3 18:15
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

Related Books

	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 Volume by Method of Cylindrical Shells

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College Calculus: Level I Volume by Method of Cylindrical Shells

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Volume by Method of Cylindrical Shells

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College Calculus 1 Online Course

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