Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Calculus Volume 2

Key Equations

Calculus Volume 2Key Equations
Menu
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

Key Equations

Area between two curves, integrating on the x-axis A=∫ab[f(x)−g(x)]dxA=∫ab[f(x)−g(x)]dx
Area between two curves, integrating on the y-axis A=∫cd[u(y)−v(y)]dyA=∫cd[u(y)−v(y)]dy
Disk Method along the x-axis V=∫abπ[f(x)]2dxV=∫abπ[f(x)]2dx
Disk Method along the y-axis V=∫cdπ[g(y)]2dyV=∫cdπ[g(y)]2dy
Washer Method V=∫abπ[(f(x))2−(g(x))2]dxV=∫abπ[(f(x))2−(g(x))2]dx
Method of Cylindrical Shells V=∫ab(2πxf(x))dxV=∫ab(2πxf(x))dx
Arc Length of a Function of x Arc Length=∫ab1+[f′(x)]2dxArc Length=∫ab1+[f′(x)]2dx
Arc Length of a Function of y Arc Length=∫cd1+[g′(y)]2dyArc Length=∫cd1+[g′(y)]2dy
Surface Area of a Function of x Surface Area=∫ab(2πf(x)1+(f′(x))2)dxSurface Area=∫ab(2πf(x)1+(f′(x))2)dx
Mass of a one-dimensional object m=∫abρ(x)dxm=∫abρ(x)dx
Mass of a circular object m=∫0r2πxρ(x)dxm=∫0r2πxρ(x)dx
Work done on an object W=∫abF(x)dxW=∫abF(x)dx
Hydrostatic force on a plate F=∫abρw(x)s(x)dxF=∫abρw(x)s(x)dx
Mass of a lamina m=ρ∫abf(x)dxm=ρ∫abf(x)dx
Moments of a lamina Mx=ρ∫ab[f(x)]22dxandMy=ρ∫abxf(x)dxMx=ρ∫ab[f(x)]22dxandMy=ρ∫abxf(x)dx
Center of mass of a lamina x–=Mymandy–=Mxmx–=Mymandy–=Mxm
Natural logarithm function lnx=∫1x1tdtlnx=∫1x1tdt Z
Exponential function y=exy=ex lny=ln(ex)=xlny=ln(ex)=x Z
PreviousNext
Order a print copy

As an Amazon Associate we earn from qualifying purchases.

Citation/Attribution

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License 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

© Jul 17, 2023 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 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.