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Table of Contents

I. Advanced Integration Techniques

Integration by Parts

24m 52s

- Intro0:00
- Important Equation0:07
- Where It Comes From (Product Rule)0:20
- Why Use It?0:35
- Lecture Example 11:24
- Lecture Example 23:30
- Shortcut: Tabular Integration7:34
- Example7:52
- Lecture Example 310:00
- Mnemonic: LIATE14:44
- Ln, Inverse, Algebra, Trigonometry, e15:38
- Additional Example 4-1
- Additional Example 5-2

Integration of Trigonometric Functions

25m 30s

- Intro0:00
- Important Equation0:07
- Powers (Odd and Even)0:19
- What To Do1:03
- Lecture Example 11:37
- Lecture Example 23:12
- Half-Angle Formulas6:16
- Both Powers Even6:31
- Lecture Example 37:06
- Lecture Example 410:59
- Additional Example 5-1
- Additional Example 6-2

Trigonometric Substitutions

30m 9s

- Intro0:00
- Important Equations0:06
- How They Work0:35
- Example1:45
- Remember: du and dx2:50
- Lecture Example 13:43
- Lecture Example 210:01
- Lecture Example 312:04
- Additional Example 4-1
- Additional Example 5-2

Partial Fractions

41m 22s

- Intro0:00
- Overview0:07
- Why Use It?0:18
- Lecture Example 11:21
- Lecture Example 26:52
- Lecture Example 313:28
- Additional Example 4-1
- Additional Example 5-2

Integration Tables

20m

- Intro0:00
- Using Tables0:09
- Match Exactly0:32
- Lecture Example 11:16
- Lecture Example 25:28
- Lecture Example 38:51
- Additional Example 4-1
- Additional Example 5-2

Trapezoidal Rule, Midpoint Rule, Left/Right Endpoint Rule

22m 36s

- Intro0:00
- Trapezoidal Rule0:13
- Graphical Representation0:20
- How They Work1:08
- Formula1:47
- Why a Trapezoid?2:53
- Lecture Example 15:10
- Midpoint Rule8:23
- Why Midpoints?8:56
- Formula9:37
- Lecture Example 211:22
- Left/Right Endpoint Rule13:54
- Left Endpoint14:08
- Right Endpoint14:39
- Lecture Example 315:32
- Additional Example 4-1
- Additional Example 5-2

Simpson's Rule

21m 8s

- Intro0:00
- Important Equation0:03
- Estimating Area0:28
- Difference from Previous Methods0:50
- General Principle1:09
- Lecture Example 13:49
- Lecture Example 26:32
- Lecture Example 39:07
- Additional Example 4-1
- Additional Example 5-2

Improper Integration

44m 18s

- Intro0:00
- Horizontal and Vertical Asymptotes0:04
- Example: Horizontal0:16
- Formal Notation0:37
- Example: Vertical1:58
- Formal Notation2:29
- Lecture Example 15:01
- Lecture Example 27:41
- Lecture Example 311:32
- Lecture Example 415:49
- Formulas to Remember18:26
- Improper Integrals18:36
- Lecture Example 521:34
- Lecture Example 6 (Hidden Discontinuities)26:51
- Additional Example 7-1
- Additional Example 8-2

II. Applications of Integrals, part 2

Arclength

23m 20s

- Intro0:00
- Important Equation0:04
- Why It Works0:49
- Common Mistake1:21
- Lecture Example 12:14
- Lecture Example 26:26
- Lecture Example 310:49
- Additional Example 4-1
- Additional Example 5-2

Surface Area of Revolution

28m 53s

- Intro0:00
- Important Equation0:05
- Surface Area0:38
- Relation to Arclength1:11
- Lecture Example 11:46
- Lecture Example 24:29
- Lecture Example 39:34
- Additional Example 4-1
- Additional Example 5-2

Hydrostatic Pressure

24m 37s

- Intro0:00
- Important Equation0:09
- Main Idea0:12
- Different Forces0:45
- Weight Density Constant1:10
- Variables (Depth and Width)2:21
- Lecture Example 13:28
- Additional Example 2-1
- Additional Example 3-2

Center of Mass

25m 39s

- Intro0:00
- Important Equation0:07
- Main Idea0:25
- Centroid1:00
- Area1:28
- Lecture Example 11:44
- Lecture Example 26:13
- Lecture Example 310:04
- Additional Example 4-1
- Additional Example 5-2

III. Parametric Functions

Parametric Curves

22m 26s

- Intro0:00
- Important Equations0:05
- Slope of Tangent Line0:30
- Arc length1:03
- Lecture Example 11:40
- Lecture Example 24:23
- Lecture Example 38:38
- Additional Example 4-1
- Additional Example 5-2

Polar Coordinates

30m 59s

- Intro0:00
- Important Equations0:05
- Polar Coordinates in Calculus0:42
- Area0:58
- Arc length1:41
- Lecture Example 12:14
- Lecture Example 24:12
- Lecture Example 310:06
- Additional Example 4-1
- Additional Example 5-2

IV. Sequences and Series

Sequences

31m 13s

- Intro0:00
- Definition and Theorem0:05
- Monotonically Increasing0:25
- Monotonically Decreasing0:40
- Monotonic0:48
- Bounded1:00
- Theorem1:11
- Lecture Example 11:31
- Lecture Example 211:06
- Lecture Example 314:03
- Additional Example 4-1
- Additional Example 5-2

Series

31m 46s

- Intro0:00
- Important Definitions0:05
- Sigma Notation0:13
- Sequence of Partial Sums0:30
- Converging to a Limit1:49
- Diverging to Infinite2:20
- Geometric Series2:40
- Common Ratio2:47
- Sum of a Geometric Series3:09
- Test for Divergence5:11
- Not for Convergence6:06
- Lecture Example 18:32
- Lecture Example 210:25
- Lecture Example 316:26
- Additional Example 4-1
- Additional Example 5-2

Integral Test

23m 26s

- Intro0:00
- Important Theorem and Definition0:05
- Three Conditions0:25
- Converging and Diverging0:51
- P-Series1:11
- Lecture Example 12:19
- Lecture Example 25:08
- Lecture Example 36:38
- Additional Example 4-1
- Additional Example 5-2

Comparison Test

22m 44s

- Intro0:00
- Important Tests0:01
- Comparison Test0:22
- Limit Comparison Test1:05
- Lecture Example 11:44
- Lecture Example 23:52
- Lecture Example 36:01
- Lecture Example 410:04
- Additional Example 5-1
- Additional Example 6-2

Alternating Series

25m 26s

- Intro0:00
- Main Theorems0:05
- Alternation Series Test (Leibniz)0:11
- How It Works0:26
- Two Conditions0:46
- Never Use for Divergence1:12
- Estimates of Sums1:50
- Lecture Example 13:19
- Lecture Example 24:46
- Lecture Example 36:28
- Additional Example 4-1
- Additional Example 5-2

Ratio Test and Root Test

33m 27s

- Intro0:00
- Theorems and Definitions0:06
- Two Common Questions0:17
- Absolutely Convergent0:45
- Conditionally Convergent1:18
- Divergent1:51
- Missing Case2:02
- Ratio Test3:07
- Root Test4:45
- Lecture Example 15:46
- Lecture Example 29:23
- Lecture Example 313:13
- Additional Example 4-1
- Additional Example 5-2

Power Series

38m 36s

- Intro0:00
- Main Definitions and Pattern0:07
- What Is The Point0:22
- Radius of Convergence Pattern0:45
- Interval of Convergence2:42
- Lecture Example 13:24
- Lecture Example 210:55
- Lecture Example 314:44
- Additional Example 4-1
- Additional Example 5-2

V. Taylor and Maclaurin Series

Taylor Series and Maclaurin Series

30m 18s

- Intro0:00
- Taylor and Maclaurin Series0:08
- Taylor Series0:12
- Maclaurin Series0:59
- Taylor Polynomial1:20
- Lecture Example 12:35
- Lecture Example 26:51
- Lecture Example 311:38
- Lecture Example 417:29
- Additional Example 5-1
- Additional Example 6-2

Taylor Polynomial Applications

50m 50s

- Intro0:00
- Main Formulas0:06
- Alternating Series Error Bound0:28
- Taylor's Remainder Theorem1:18
- Lecture Example 13:09
- Lecture Example 29:08
- Lecture Example 317:35
- Additional Example 4-1
- Additional Example 5-2

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

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

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1 answer

Last reply by: Dr. William Murray

Mon Oct 21, 2013 10:31 PM

Post by Wael Saleh on October 20, 2013

Great job, As you know there are many types of tests, Therefore, I have problem !! Are there any certain signals in question describe which kind of tests should I use it to solve problem?

Best regards,

1 answer

Last reply by: Dr. William Murray

Sun Apr 28, 2013 11:12 AM

Post by Megan Kell on April 10, 2012

I'm so confused... I understand the p-series method and the integral method, but what do they have to do to each other? and at the beginning of this lecture, he talks about the three points you must identify before doing the integral test, but yet he only covers that in additional example IV, although he should have demonstrated this in Example 3 as well..? The thought process in this lecture is not very clearly executed.

1 answer

Last reply by: Dr. William Murray

Sun Apr 28, 2013 11:04 AM

Post by Jaspreet Singh on March 15, 2010

There's a typo in the notes, for number 3 it should be "Is f'(x) negative"

1 answer

Last reply by: Dr. William Murray

Sun Apr 28, 2013 10:44 AM

Post by John Haddad on February 10, 2010

Tabular integration is a great technique for Ex 3, unfortunately my university disallows it's use on exams; but it'll come in handy for multiple choice problems.