Problem Details

Calculus: Early Transcendentals 6th Edition

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Chapter 1 Chapter 1: FUNCTIONS AND MODELS

Section 1 Section 1: Four Ways to Represent a Function

Section 2 Section 2: Mathematical Models: A Catalog of Essential Functions

Section 3 Section 3: New Functions from Old Functions

Section 4 Section 4: Graphing Calculators and Computers

Section 5 Section 5: Exponential Functions

Section 6 Section 6: Inverse Functions and Logarithms

Section 7 Section 7: Review

Chapter 2 Chapter 2: LIMITS AND DERIVATIVES

Section 1 Section 1: The Tangent and Velocity Problems

Section 2 Section 2: The Limit of a Function

Section 3 Section 3: Calculating Limits Using the Limit Laws

Section 4 Section 4: The Precise Definition of a Limit

Section 5 Section 5: Continuity

Section 6 Section 6: Limits at Infinity; Horizontal Asymptotes

Section 7 Section 7: Derivatives and Rates of Change

Section 8 Section 8: The Derivative as a Function

Chapter 3 Chapter 3: DIFFERENTIATION RULES

Section 1 Section 1: Derivatives of Polynomials and Exponential Functions

Section 2 Section 2: The Product and Quotient Rules

Section 3 Section 3: Derivatives of Trigonometric Functions

Section 4 Section 4: The Chain Rule

Section 5 Section 5: Implicit Differentiation

Section 6 Section 6: Derivatives of Logarithmic Functions

Section 7 Section 7: Rates of Change in the Natural and Social Sciences

Section 8 Section 8: Exponential Growth and Decay

Section 9 Section 9: Related Rates

Section 10 Section 10: Linear Approximations and Differentials

Section 11 Section 11: Hyperbolic Functions

Chapter 4 Chapter 4: APPLICATIONS OF DIFFERENTIATION

Section 1 Section 1: Maximum and Minimum Values

Section 2 Section 2: The Mean Value Theorem

Section 3 Section 3: How Derivatives Affect the Shape of a Graph

Section 4 Section 4: Indeterminate Forms and L'Hospital's Rule

Section 5 Section 5: Summary of Curve Sketching

Section 6 Section 6: Graphing with Calculus andCalculators

Section 7 Section 7: Optimization Problems

Section 8 Section 8: Newton's Method

Section 9 Section 9: Antiderivatives

Chapter 5 Chapter 5: INTEGRALS

Section 1 Section 1: Areas and Distances

Section 2 Section 2: The Definite Integral

Section 3 Section 3: The Fundamental Theorem of Calculus

Section 4 Section 4: Indefinite Integrals and the Net Change Theorem

Section 5 Section 5: The Substitution Rule

Chapter 6 Chapter 6: INTEGRALS

Section 1 Section 1: Areas between Curves

Section 2 Section 2: Volumes

Section 3 Section 3: Volumes by Cylindrical Shells

Section 4 Section 4: Work

Section 5 Section 5: Average Value of a Function

Chapter 7 Chapter 7: TECHNIQUES OF INTEGRATION

Section 1 Section 1: Integration by Parts

Section 2 Section 2: Trigonometric Integrals

Section 3 Section 3: Trigonometric Substitution

Section 4 Section 4: Integration of Rational Functions by Partial Fractions

Section 5 Section 5: Strategy for Integration

Section 6 Section 6: Integration Using Tables and Computer Algebra Systems

Section 7 Section 7: Approximate Integration

Section 8 Section 8: Improper Integrals

Chapter 8 Chapter 8: FURTHER APPLICATIONS OF INTEGRATION

Section 1 Section 1: Arc Length

Section 2 Section 2: Area of a Surface of Revolution

Section 3 Section 3: Applications to Physics and Engineering

Section 4 Section 4: Applications to Economics and Biology

Section 5 Section 5: Probability

Chapter 9 Chapter 9: DIFFERENTIAL EQUATIONS

Section 1 Section 1: Modeling with Differential Equations

Section 2 Section 2: Direction Fields and Euler's Method

Section 3 Section 3: Separable Equations

Section 4 Section 4: Models for Population Growth

Section 5 Section 5: Linear Equations

Section 6 Section 6: Predator-Prey Systems

Chapter 10 Chapter 10: PARAMETRIC EQUATIONS AND POLAR COORDINATES

Section 1 Section 1: Curves Defined by Parametric Equations

Section 2 Section 2: Calculus with Parametric Curves

Section 3 Section 3: Polar Coordinates

Section 4 Section 4: Areas and Lengths in Polar Coordinates

Section 5 Section 5: Conic Sections

Section 6 Section 6: Conic Sections in Polar Coordinates

Section 7 Section 7: Review

Chapter 11 Chapter 11: INFINITE SEQUENCES AND SERIES

Section 1 Section 1: Sequences

Section 2 Section 2: Series

Section 3 Section 3: The Integral Test and Estimates of Sums

Section 4 Section 4: The Comparison Tests

Section 5 Section 5: Alternating Series

Section 6 Section 6: Absolute Convergence and the Ratio and Root Tests

Section 7 Section 7: Strategy for Testing Series

Section 8 Section 8: Power Series

Section 9 Section 9: Representations of Functions as Power Series

Section 10 Section 10: Taylor and Maclaurin Series

Section 11 Section 11: Applications of Taylor Polynomials

Section 13 Section 13: Review

Chapter 12 Chapter 12: VECTORS AND THE GEOMETRY OF SPACE

Section 1 Section 1: Three-Dimensional Coordinate Systems

Section 2 Section 2: Vectors

Section 3 Section 3: The Dot Product

Section 4 Section 4: The Cross Product

Section 5 Section 5: Equations of Lines and Planes

Section 6 Section 6: Cylinders and Quadric Surfaces

Chapter 13 Chapter 13: VECTOR FUNCTIONS

Section 1 Section 1: Vector Functions and Space Curves

Section 2 Section 2: Derivatives and Integrals of Vector Functions

Section 3 Section 3: Arc Length and Curvature

Section 4 Section 4: Motion in Space: Velocity and Acceleration

Chapter 14 Chapter 14: PARTIAL DERIVATIVES

Section 1 Section 1: Functions of Several Variables

Section 2 Section 2: Limits and Continuity

Section 3 Section 3: Partial Derivatives

Section 4 Section 4: Tangent Planes and Linear Approximations

Section 5 Section 5: The Chain Rule

Section 6 Section 6: Directional Derivatives and the Gradient Vector

Section 7 Section 7: Maximum and Minimum Values

Section 8 Section 8: Lagrange Multipliers

Chapter 15 Chapter 15: MULTIPLE INTEGRALS

Section 1 Section 1: Double Integrals over Rectangles

Section 2 Section 2: Iterated Integrals

Section 3 Section 3: Double Integrals over General Regions

Section 4 Section 4: Double Integrals in Polar Coordinates

Section 5 Section 5: Applications of Double Integrals

Section 6 Section 6: Triple Integrals

Section 7 Section 7: Triple Integrals in Cylindrical Coordinates

Section 8 Section 8: Triple Integrals in Spherical Coordinates

Section 9 Section 9: Change of Variables in Multiple Integrals

Chapter 16 Chapter 16: VECTOR CALCULUS

Section 1 Section 1: Vector Fields

Section 2 Section 2: Line Integrals

Section 3 Section 3: The Fundamental Theorem for Line Integrals

Section 4 Section 4: Green's Theorem

Section 5 Section 5: Curl and Divergence

Section 6 Section 6: Parametric Surfaces and Their Areas

Section 7 Section 7: Surface Integrals

Section 8 Section 8: Stokes' Theorem

Section 9 Section 9: The Divergence Theorem

Section 10 Section 10: Summary

Chapter 17 Chapter 17: SECOND-ORDER DIFFERENTIAL EQUATIONS

Section 1 Section 1: Second-Order Linear Equations

Section 2 Section 2: Nonhomogeneous Linear Equations

Section 3 Section 3: Applications of Second-Order Differential Equations

Section 4 Section 4: Series Solutions

Section 5 Section 5: Review

Problem 1.22

Report 30 1

A spherical balloon with radius $r$ inches has volume
$$V(r) =\displaystyle \frac{4}{3}\pi r^{3}$$. Find a function that represents the amount of air
required to inflate the balloon from a radius of $r$ inches to a
radius of $r+ 1$ inches.

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