  Professor Switkes

Volume by Method of Disks and Washers

Slide Duration:

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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
-1
-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
5:31
Lecture Example 1
6:05
Lecture Example 2
8:28
Lecture Example 3
11:55
-1
-2
Volume by Method of Cylindrical Shells

30m 29s

Intro
0:00
Important Equations
0:50
1:04
Equation 2: Rotation about y-axis (2 curves)
7:34
8:15
Lecture Example 1
8:57
Lecture Example 2
14:26
Lecture Example 3
18:15
-1
-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
-1
-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
-1
-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
-1
-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
-1
-2
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• ## Related Books 0 answersPost by Richard Gregory on October 17, 2013Hi. I worked example three by converting the graph and co-ordinates into x values and then subtracting the area under the curve from the total disc volume. Is this a valid method? The answer was the same. 0 answersPost by Kimberly McDevitt on November 19, 2012There is something wrong with example number 4. The video jams around 1:50 and plays like a broken record. Please fix. Thank you. 0 answersPost by Daniela Valencia on April 17, 2012The lecture examples are very simple for a college calculus class. No very helpful. 1 answerLast reply by: Mohanraj PonrajSat Jan 19, 2013 10:45 AMPost by Matthew Odunjo on June 21, 2011please I didnt why the first Big radius is 4 and the small being x 2 answersLast reply by: Mary ShriverSat May 26, 2012 9:54 AMPost by Ruben Ayllon Ayllon on April 11, 2011In Lecture Example 1 isn't the answer 128/7 pie. If not could you explain how you got the answer? 0 answersPost by Fatima Sulaiman on May 4, 2010how rude. These videos are pretty good except there needs to be 3d computerized examples for the area and volume problems.

### Volume by Method of Disks and Washers

• 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 Disks and Washers

Find the volume of the solid that results from revolving y = x from x = 0 to x = 4 around the x-axis.
• • The area of a slice would be πr2. The volume would be πr2 * thickness
• So the volume of a tiny slice would be πr2 dx where dx is the thickness
• In this problem, we're rotating around the x-axis, so the radius is just y in this case.
• Volume of slice = πy2 dx = πx2 dx
• Total volume from x = 0 to x = 4 is V
• V = ∫04 πx2 dx
• V = π[(x3)/3] |04
• V = π[64/3] − 0
• We just found the volume of a cone using calculus!
V = [(64 π)/3]
Find the volume of the solid that results from revolving x = 4 from y = 0 to y = 6 around the y-axis.
• • V = π∫06 42 dy
• V = π16x |06
• V = 96π− 0
• Now we found the volume of a cylinder.
V = 96π
Find the volume of the solid that results from revolving y = secx from x = −[(π)/4] to x = [(π)/4] around the x-axis.
• • V = π∫−[(π)/4][(π)/4] sec2 x  dx
• V = πtanx |−[(π)/4][(π)/4]
• V = π(tan[(π)/4] − tan[(−π)/4])
• V = π(1 − (−1))
V = 2π
Find the volume of the solid that results from revolving y = lnx from y = 0 to y = 1 about the y-axis.
• • V = π∫01 x2 dy
• y = lnx
• ey = elnx
• x = ey
• V = π∫01 e2y dy
• u = 2y
• du = 2  dy
• V = [(π)/2] ∫01 eu du
• V = [(π)/2] eu |01
• V = [(π)/2] e2y |01
• V = [(π)/2] (e2 − e0)
V = [(π)/2] (e2 − 1)
Find the volume of the solid that results from revolving y = [lnx/(√x)] from x = 1 to x = e about the x-axis.
• • V = π∫1e [((lnx)2)/x] dx
• u = lnx
• du = [1/x] dx
• V = π∫1e u2 du
• V = π[(u3)/3] |1e
• V = π[((lnx)3)/3] |1e
• V = [(π)/3] ( (lne)3 − (ln1)3 )
V = [(π)/3]
Find the volume of the solid that results from revolving The area bounded by y = ex, y = 1, and x = 2 around the x-axis.
• • V = π∫02 (ex)2 − 12 dx
• V = π∫02 e2x − 1  dx
• V = π([1/2] e2x − x) |02
• V = π([1/2] e4 − 2 − [1/2] e0 + 0)
• V = π([1/2] e4 − [5/2])
V = [(π)/2] (e4 − 5)
Find the volume of the solid that results from revolving y = x2 + 1 from x = 0 to x = 1 around the x-axis.
• • V = π∫01 (x2 + 1)2 dx
• V = π∫01 x4 + 2x2 + 1  dx
• V = π([(x5)/5] + [(2x3)/3] + x) |01
• V = π([1/5] + [2/3] + 1 − 0 − 0 − 0)
V = [(28π)/15]
Find the volume of the solid that results from revolving y = √{4 − x2} from x = −2 to x = 2 around the x-axis.
• • V = π∫−22 (√{4 − x2})2 dx
• V = π∫−22 4 − x2 dx
• V = π(4x − [(x3)/3]) |−22
• V = π(8 − [8/3] − (−8 + [8/3]))
• V = π(16 − [16/3])
• This is the volume of a sphere of radius 2!
V = [(32π)/3]
Find the volume of the solid that results from revolving y = [1/(√x)] from x = 1 to x = 3 around the x-axis.
• • V = π∫13 [1/x] dx
• V = πlnx |13
• V = π(ln3 − ln1)
• V = π(ln3 − 0)
V = πln3
The area bounded by y = x3 and y = x is rotated about the x-axis. Find the volume of the resulting solid.
• • Find their points of intersection
• x3 = x
• x = 0, 1, −1
• V = π(− ∫−10 (x)2 − (x3)2 dx) + π∫01 (x)2 − (x3)2 dx
• We can use symmetry here instead of using two separate integrals
• V = 2 π∫01 (x)2 − (x3)2 dx
• V = 2 π([(x3)/3] − [(x7)/7]) |01
• V = 2π([1/3] − [1/7] − 0)
V = [(8π)/21]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Volume by Method of Disks and Washers

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
• Important Equations 0:16
• Equation 1: Rotation about x-axis (disks)
• Equation 2: Two curves about x-axis (washers)
• Equation 3: Rotation about y-axis
• Lecture Example 1 6:05
• Lecture Example 2 8:28
• Lecture Example 3 11:55

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