MA442

Whan Ki Lee, Jan 3, 2017

Calculus II

Table of Contents

  1. Chapter 5. Applications of Integration
    1. Upper and lower Riemann Sums
    2. Volume of Solids of Revolution
    3. Solid of Revolution
    4. Visualizing the volume of revolution - cylindrical shells
    5. Volume - Shell Method
    6. Visual Calculus - Volumes and Areas
  2. Chapter 6. Inverse Functions
    1. Inverse Function and Derivative
  3. Chapter 7. Techniques of Integration
    1. Improper Integrals
    2. Comparison Test for Improper Integrals
  4. Chapter 8. Further Applications of Integration
    1. Calculus - Arc Length
    2. Copy of surface area of a solid of revolution
  5. Chapter 11. Infinite Sequences and Series
    1. The Integral Test
    2. Integral Test - Divergent Series
    3. Taylor Series
    4. Taylor polynomial graphs

1.

Chapter 5. Applications of Integration

5.2 Volumes, 5.3 Volumes by Cylindrical Shells.

  • 1. Upper and lower Riemann Sums
  • 2. Volume of Solids of Revolution
  • 3. Solid of Revolution
  • 4. Visualizing the volume of revolution - cylindrical shells
  • 5. Volume - Shell Method
  • 6. Visual Calculus - Volumes and Areas

1.1.

Upper and lower Riemann Sums

This applet shows how upper and lower Riemann sums can approximate an integral [math]\int^b_a f(x) \, \textrm{d}x[/math][br]Further, they show that as the number of strips [math]n[/math] increases, the Riemann sums converge to true value of the definite integral.[br]Input your own function into the textbox and set the limits to different values.
– Dr Adrian Jannetta

1.2.

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1.3.

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1.4.

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1.5.

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1.6.

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2.

Chapter 6. Inverse Functions

6.1 Inverse Functions, 6.2* The Natural Logarithmic Function, 6.3* The Natural Exponential Function, 6.4* General Logarithmic and Exponential Functions, 6.5 Exponential Growth and Decay, 6.6 Inverse Trigonometric Functions, 6.8 Indeterminate Forms and L’Hospital’s Rule

  • 1. Inverse Function and Derivative

2.1.

Inverse Function and Derivative

– Whan Ki Lee

3.

Chapter 7. Techniques of Integration

7.1 Integration by Parts, 7.2 Trigonometric Integrals, 7.3 Trigonometric Substitution, 7.4 Integration of Rational Functions by Partial Fraction,s 7.5 Strategy for Integration, 7.6 Integration Using Tables and Computer Algebra Systems, 7.8 Improper Integrals.

  • 1. Improper Integrals
  • 2. Comparison Test for Improper Integrals

3.1.

Improper Integrals

– Whan Ki Lee

3.2.

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4.

Chapter 8. Further Applications of Integration

8.1 Arc Length. 8.2 Area of a Surface of Revolution.

  • 1. Calculus - Arc Length
  • 2. Copy of surface area of a solid of revolution

4.1.

Calculus - Arc Length

– Chip Rollinson

4.2.

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5.

Chapter 11. Infinite Sequences and Series

11.1 Sequences, 11.2 Series, 11.3 The Integral Test, (pp. 738-741) 11.4 The Comparison Tests, 11.5 Alternating Series, 11.6 Absolute Convergence and the Ratio and Root Tests, 11.7 Strategy for Testing Series, 11.8 Power Series, 11.9 Representations of Functions as Power Series, 11.10 Taylor and Maclaurin Series, 11.11 Applications of Taylor Polynomials.

  • 1. The Integral Test
  • 2. Integral Test - Divergent Series
  • 3. Taylor Series
  • 4. Taylor polynomial graphs

5.1.

The Integral Test

Consider the series Σ1/n². How does the area of the rectangles plotted below compare with the sum of the series?
Click the checkbox to show the function f(x) = 1/x². Notice that the sum of the areas of the boxes is the sum of the series - but it is also a left Riemann sum for the integral of f(x) from 1 to ∞ (with one box leftover). If you drag the slider over, you can also see that the areas of the boxes make a right Riemann sum for the integral of f(x) from 1 to ∞ (without any leftover). So if the integral makes sense, so does the series!
– Jason McCullough

5.2.

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5.3.

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5.4.

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