Book cover for Thomas Calculus

Thomas Calculus

George B. Thomas, Jr.

ISBN #9780321878960

13th Edition

6,812 Questions

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127,035 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section extends the concept of definite integration beyond finding areas under curves to calculating volumes of solids by slicing, using methods such as the disk, washer, and cylindrical shell techniques. By identifying the relevant cross-sectional areas and integrating over the appropriate interval, one can derive formulas for the volume of diverse geometric shapes and revolution solids. Mastery of these methods is essential for solving practical problems encountered in science, engineering, and further mathematics.

Learning Objectives

1

Explain how to use definite integrals to compute the volume of solids through cross-sectional slicing.

2

Apply the disk, washer, and cylindrical shell methods to set up and evaluate volume integrals.

3

Translate geometric descriptions of solids into appropriate integrals by identifying cross-sectional areas.

4

Solve real-world problems involving volumes, including applications in physics and engineering.

Key Concepts

CONCEPT

DEFINITION

Definite Integral

A limit of Riemann sums used to calculate the accumulation of quantities, such as area and volume, over an interval.

Cross-Section

The intersection of a solid with a plane; its area, A(x), is used in integration to determine volume.

Disk Method

A technique to calculate volumes of solids of revolution where cross-sections perpendicular to the axis of rotation are disks with area A(x) = π[R(x)]².

Washer Method

An extension of the disk method used when the solid of revolution has a hole, with cross-sectional area A(x) = π([R(x)]² - [r(x)]²), where R(x) and r(x) are the outer and inner radii.

Cylindrical Shell Method

A method of volume calculation obtained by integrating the lateral surface area of thin cylindrical shells, useful when slicing vertically instead of horizontally.

Example Problems

Example 1

The solid lies between planes perpendicular to the $x$ -axis at $x=0$ and $x=4 .$ The cross-sections perpendicular to the axis on the interval $0 \leq x \leq 4$ are squares whose diagonals run from the parabola $y=-\sqrt{x}$ to the parabola $y=\sqrt{x}$ .

Example 2

The solid lies between planes perpendicular to the $x$ -axis at $x=-1$ and $x=1 .$ The cross-sections perpendicular to the $x$ -axis are circular disks whose diameters run from the parabola $y=x^{2}$ to the parabola $y=2-x^{2}$

Example 3

The solid lies between planes perpendicular to the $x$ -axis at $x=-1$ and $x=1 .$ The cross-sections perpendicular to the $x$ -axis between these planes are squares whose bases run from the semicircle $y=-\sqrt{1-x^{2}}$ to the semicircle $y=\sqrt{1-x^{2}}$

Example 4

The solid lies between planes perpendicular to the $x$ -axis at $x=-1$ and $x=1 .$ The cross-sections perpendicular to the $x$ -axis between these planes are squares whose diagonals run from the semicircle $y=-\sqrt{1-x^{2}}$ to the semicircle $y=\sqrt{1-x^{2}}$

Example 5

The base of a solid is the region between the curve $y=2 \sqrt{\sin x}$ and the interval $[0, \pi]$ on the $x$ -axis. The cross-sections perpendicular to the $x$ -axis are a. equilateral triangles with bases running from the $x$ -axis to the curve as shown in the accompanying figure. b. squares with bases running from the $x$ -axis to the curve.
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Step-by-Step Explanations

QUESTION

How do you compute the volume of a solid with known cross-sectional area A(x)?

STEP-BY-STEP ANSWER:

Step 1: Sketch the solid and identify the orientation and shape of the cross-sections.
Step 2: Determine the formula for the cross-sectional area A(x) as a function of the position x.
Step 3: Identify the interval [a, b] over which the cross-sections occur.
Step 4: Set up the integral V = ∫[a,b] A(x) dx to represent the total volume.
Final Answer: V = ∫[a,b] A(x) dx.

Volume Using Cross-Sections (Slicing Method)

QUESTION

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

STEP-BY-STEP ANSWER:

Step 1: Determine the radius R(x) from the axis of rotation to the curve.
Step 2: Write the area of a typical disk as A(x) = π [R(x)]².
Step 3: Identify the correct limits of integration along the axis of revolution.
Step 4: Integrate using V = ∫[a,b] π [R(x)]² dx.
Final Answer: V = ∫[a,b] π [R(x)]² dx.

Disk Method

QUESTION

How do you calculate the volume of a solid with a cavity using the washer method?

STEP-BY-STEP ANSWER:

Step 1: Identify the outer radius R(x) and inner radius r(x) for the washers.
Step 2: Compute the area of a typical washer as A(x) = π([R(x)]² - [r(x)]²).
Step 3: Establish the limits of integration based on the geometry.
Step 4: Set up the volume integral V = ∫[a,b] π([R(x)]² - [r(x)]²) dx and evaluate.
Final Answer: V = ∫[a,b] π([R(x)]² - [r(x)]²) dx.

Washer Method

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Common Mistakes

  • Using incorrect limits of integration or misidentifying the bounds over which the cross-sections exist.
  • Mixing up the formulas for different methods, such as not squaring the radius in the disk method or forgetting to subtract the inner area in the washer method.
  • Failing to correctly visualize the solid and its cross-sectional shapes, which can lead to an incorrect expression for A(x).
  • Neglecting to simplify the integrand properly before integration, resulting in algebraic errors.