Book cover for Calculus of a Single Variable

Calculus of a Single Variable

Ron Larson, Bruce Edwards

ISBN #9781285060286

10th Edition

6,214 Questions

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101,749 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section emphasizes the application of definite integrals to compute the area between two curves and the volume of solids of revolution. Key techniques include partitioning the region, using representative rectangles, and establishing proper integrals. Importantly, proper identification of the upper and lower functions or outer and inner radii is crucial. By interpreting integration as an accumulation process, students gain powerful tools for solving geometric problems in both two and three dimensions.

Learning Objectives

1

Describe how integration can be used as an accumulation process to determine areas between curves.

2

Set up and evaluate definite integrals for regions bounded by two curves, including situations with intersecting curves.

3

Explain and apply the disk and washer methods to find the volume of solids of revolution.

4

Analyze representative elements (vertical and horizontal rectangles) to structure integration in various applications.

Key Concepts

CONCEPT

DEFINITION

Definite Integral

An expression representing the accumulation (or net area) under a curve between two bounds.

Area Between Two Curves

The area of the region bounded vertically by two continuous functions, calculated as the integral of the difference between the upper and lower functions over a specified interval.

Representative Rectangle

A small element or slice of a region with a specified width (Δx or Δy) whose area approximates the area of a part of the region; used in forming the Riemann Sum.

Solid of Revolution

A three-dimensional figure obtained by revolving a plane region about a line (the axis of revolution).

Disk Method

A technique for computing the volume of a solid of revolution by approximating the solid with a series of disks; the volume of each disk is determined using its radius and thickness.

Washer Method

An extension of the disk method used when the solid of revolution has a hole; the volume is determined by subtracting the inner disk area from the outer disk area.

Example Problems

Example 1

Writing a Definite Integral In Exercises $1-6,$ set up the definite integral that gives the area of the region. $$ \begin{array}{l}{y_{1}=x^{2}-6 x} \\ {y_{2}=0}\end{array} $$

Example 2

Writing a Definite Integral In Exercises $1-6,$ set up the definite integral that gives the area of the region. $$ \begin{array}{l}{y_{1}=x^{2}+2 x+1} \\ {y_{2}=2 x+5}\end{array} $$

Example 3

Writing a Definite Integral In Exercises $1-6,$ set up the definite integral that gives the area of the region. $$ \begin{array}{l}{y_{1}=x^{2}-4 x+3} \\ {y_{2}=-x^{2}+2 x+3}\end{array} $$

Example 4

Writing a Definite Integral In Exercises $1-6,$ set up the definite integral that gives the area of the region. $$ \begin{array}{l}{y_{1}=x^{2}} \\ {y_{2}=x^{3}}\end{array} $$

Example 5

Writing a Definite Integral In Exercises $1-6,$ set up the definite integral that gives the area of the region. $$ \begin{array}{l}{y_{1}=3\left(x^{3}-x\right)} \\ {y_{2}=0}\end{array} $$
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Step-by-Step Explanations

QUESTION

How do you compute the area between two curves f(x) and g(x) on the interval [a, b]?

STEP-BY-STEP ANSWER:

Step 1: Identify the functions f(x) (the upper function) and g(x) (the lower function) over the interval [a, b].
Step 2: Partition the interval [a, b] into n subintervals of equal width Δx.
Step 3: In each subinterval, construct a representative rectangle with height equal to f(x_i) - g(x_i) and width Δx.
Step 4: Sum the areas of these rectangles to form the Riemann sum: Σ [f(x_i) - g(x_i)] Δx.
Step 5: Take the limit as n approaches infinity (Δx → 0) to obtain the definite integral: Area = ∫[a to b] (f(x) - g(x)) dx.
Final Answer: The area between the curves is given by ∫[a,b] [f(x) - g(x)] dx.

Area Between Curves

QUESTION

How can you find the volume of a solid of revolution using the disk method?

STEP-BY-STEP ANSWER:

Step 1: Identify the region to be revolved and the axis of revolution.
Step 2: Determine the radius R(x) of a representative disk at a typical point x; this is the distance from the axis of revolution to the function.
Step 3: Express the volume of a thin disk as dV = π [R(x)]² Δx.
Step 4: Sum the volumes of all disks by setting up the integral: Volume = ∫[a to b] π [R(x)]² dx.
Step 5: Evaluate the definite integral to obtain the total volume.
Final Answer: The volume is given by V = π ∫[a,b] [R(x)]² dx.

Volume Using the Disk Method

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Common Mistakes

  • Failing to correctly identify which function lies on top (or outside) resulting in negative or zero area.
  • Incorrectly setting up the limits of integration, especially when curves intersect or boundaries change.
  • Omitting the subtraction step in the area between curves (i.e., not subtracting the lower function from the upper function).
  • For the disk method, not squaring the function to compute the area of the disk or misidentifying the radius.