Skip to Content
OpenStax Logo
Calculus Volume 2

Chapter Review Exercises

Calculus Volume 2Chapter Review Exercises
Buy book
  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. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter 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. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter 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. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter 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. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter 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. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Chapter 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. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter 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. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter 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

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(ba).F(ba).

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=x2x,y=x2x, 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=2y2y3,x=0,andy=0x=2y2y3,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=x3,x=0,y=0,andx=2y=x3,x=0,y=0,andx=2

444.

y=x2xandx=0y=x2xandx=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=1xy=1x

449.

Find the mass of ρ=exρ=ex 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=3yx=3y 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=3x,y=3x, 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.

For the following exercises, consider the Karun-3 dam in Iran. Its shape can be approximated as an isosceles triangle with height 205205 m and width 388388 m. Assume the current depth of the water is 180180 m. The density of water is 10001000 kg/m 3.3.

456.

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.

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

Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution-NonCommercial-ShareAlike License 4.0 and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction
Citation information

© Mar 30, 2016 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.