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Calculus Volume 2

Review Exercises

Calculus Volume 2Review Exercises
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Table of contents
  1. Preface
  2. 1 Integration
    1. Introduction
    2. 1.1 Approximating Areas
    3. 1.2 The Definite Integral
    4. 1.3 The Fundamental Theorem of Calculus
    5. 1.4 Integration Formulas and the Net Change Theorem
    6. 1.5 Substitution
    7. 1.6 Integrals Involving Exponential and Logarithmic Functions
    8. 1.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  3. 2 Applications of Integration
    1. Introduction
    2. 2.1 Areas between Curves
    3. 2.2 Determining Volumes by Slicing
    4. 2.3 Volumes of Revolution: Cylindrical Shells
    5. 2.4 Arc Length of a Curve and Surface Area
    6. 2.5 Physical Applications
    7. 2.6 Moments and Centers of Mass
    8. 2.7 Integrals, Exponential Functions, and Logarithms
    9. 2.8 Exponential Growth and Decay
    10. 2.9 Calculus of the Hyperbolic Functions
    11. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  4. 3 Techniques of Integration
    1. Introduction
    2. 3.1 Integration by Parts
    3. 3.2 Trigonometric Integrals
    4. 3.3 Trigonometric Substitution
    5. 3.4 Partial Fractions
    6. 3.5 Other Strategies for Integration
    7. 3.6 Numerical Integration
    8. 3.7 Improper Integrals
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  5. 4 Introduction to Differential Equations
    1. Introduction
    2. 4.1 Basics of Differential Equations
    3. 4.2 Direction Fields and Numerical Methods
    4. 4.3 Separable Equations
    5. 4.4 The Logistic Equation
    6. 4.5 First-order Linear Equations
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  6. 5 Sequences and Series
    1. Introduction
    2. 5.1 Sequences
    3. 5.2 Infinite Series
    4. 5.3 The Divergence and Integral Tests
    5. 5.4 Comparison Tests
    6. 5.5 Alternating Series
    7. 5.6 Ratio and Root Tests
    8. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  7. 6 Power Series
    1. Introduction
    2. 6.1 Power Series and Functions
    3. 6.2 Properties of Power Series
    4. 6.3 Taylor and Maclaurin Series
    5. 6.4 Working with Taylor Series
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  8. 7 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 7.1 Parametric Equations
    3. 7.2 Calculus of Parametric Curves
    4. 7.3 Polar Coordinates
    5. 7.4 Area and Arc Length in Polar Coordinates
    6. 7.5 Conic Sections
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index

Review Exercises

True or False? Justify your answer with a proof or a counterexample.

435.

The amount of work to pump the water out of a half-full cylinder is half the amount of work to pump the water out of the full cylinder.

436.

If the force is constant, the amount of work to move an object from x=ax=a to x=bx=b is F(b−a).F(b−a).

437.

The disk method can be used in any situation in which the washer method is successful at finding the volume of a solid of revolution.

438.

If the half-life of seaborgium-266seaborgium-266 is 360360 ms, then k=(ln(2))/360.k=(ln(2))/360.

For the following exercises, use the requested method to determine the volume of the solid.

439.

The volume that has a base of the ellipse x2/4+y2/9=1x2/4+y2/9=1 and cross-sections of an equilateral triangle perpendicular to the y-axis.y-axis. Use the method of slicing.

440.

y=x2−x,y=x2−x, from x=1tox=4,x=1tox=4, rotated around they-axis using the washer method

441.

x=y2x=y2 and x=3yx=3y rotated around the y-axis using the washer method

442.

x=2y2−y3,x=0,andy=0x=2y2−y3,x=0,andy=0 rotated around the x-axis using cylindrical shells

For the following exercises, find

  1. the area of the region,
  2. the volume of the solid when rotated around the x-axis, and
  3. the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.
443.

y = x 3 , x = 0 , y = 0 , and x = 2 y = x 3 , x = 0 , y = 0 , and x = 2

444.

y = x 2 − x and x = 0 y = x 2 − x and x = 0

445.

[T] y=ln(x)+2andy=xy=ln(x)+2andy=x

446.

y=x2y=x2 and y=xy=x

447.

y=5+x,y=5+x, y=x2,y=x2, x=0,x=0, and x=1x=1

448.

Below x2+y2=1x2+y2=1 and above y=1−xy=1−x

449.

Find the mass of ρ=e−xρ=e−x on a disk centered at the origin with radius 4.4.

450.

Find the center of mass for ρ=tan2xρ=tan2x on x∈(−π4,π4).x∈(−π4,π4).

451.

Find the mass and the center of mass of ρ=1ρ=1 on the region bounded by y=x5y=x5 and y=x.y=x.

For the following exercises, find the requested arc lengths.

452.

The length of xx for y=cosh(x)y=cosh(x) from x=0tox=2.x=0tox=2.

453.

The length of yy for x=3−yx=3−y from y=0y=0 to y=4y=4

For the following exercises, find the surface area and volume when the given curves are revolved around the specified axis.

454.

The shape created by revolving the region between y=4+x,y=4+x, y=3−x,y=3−x, x=0,x=0, and x=2x=2 rotated around the y-axis.

455.

The loudspeaker created by revolving y=1/xy=1/x from x=1x=1 to x=4x=4 around the x-axis.

456.

For this exercise, consider the Karun-3 dam in Iran. Its shape can be approximated as an inverted isosceles triangle spanning across the river, with height 205 m and width (across the top of the dam) 388 m. Assume the current depth of the water is 180 m. The density of water is 1000 kg/m3. Find the total force on the wall of the dam.

457.

You are a crime scene investigator attempting to determine the time of death of a victim. It is noon and 45°F45°F outside and the temperature of the body is 78°F.78°F. You know the cooling constant is k=0.00824°F/min.k=0.00824°F/min. When did the victim die, assuming that a human’s temperature is 98°F98°F ?

For the following exercise, consider the stock market crash in 19291929 in the United States. The table lists the Dow Jones industrial average per year leading up to the crash.

Years after 1920 Value ($)
11 63.9063.90
33 100100
55 110110
77 160160
99 381.17381.17
Source: http://stockcharts.com/freecharts/historical/djia19201940.html
458.

[T] The best-fit exponential curve to these data is given by y=40.71+1.224x.y=40.71+1.224x. Why do you think the gains of the market were unsustainable? Use first and second derivatives to help justify your answer. What would this model predict the Dow Jones industrial average to be in 20142014 ?

For the following exercises, consider the catenoid, the only solid of revolution that has a minimal surface, or zero mean curvature. A catenoid in nature can be found when stretching soap between two rings.

459.

Find the volume of the catenoid y=cosh(x)y=cosh(x) from x=−1tox=1x=−1tox=1 that is created by rotating this curve around the x-axis,x-axis, as shown here.

This figure is an image of a catenoid. It has been formed by rotating a catenary curve about a vertical axis.
460.

Find surface area of the catenoid y=cosh(x)y=cosh(x) from x=−1x=−1 to x=1x=1 that is created by rotating this curve around the x-axis.x-axis.

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