Textbook Question
14–25. {Use of Tech} Areas of regions Determine the area of the given region.
The region in the first quadrant bounded by y = x/6 and y = 1−|x/2−1|
Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 6, Problem 6.R.34b
Area and volume The region R is bounded by the curves x = y²+2,y=x−4, and y=0 (see figure).
b. Write a single integral that gives the volume of the solid generated when R is revolved about the x-axis.
Verified step by step guidance1
First, identify the region R bounded by the curves: \( x = y^2 + 2 \), \( y = x - 4 \), and \( y = 0 \). Sketching or visualizing these curves helps understand the limits of integration and the shape of the region.
Since the solid is generated by revolving the region R about the x-axis, consider using the method of cylindrical shells or washers/disks. Here, using the washer method with respect to \( y \) is convenient because the boundaries are given in terms of \( y \) and \( x \).
Express the volume element as a washer with inner radius and outer radius measured from the x-axis. The outer radius is the distance from the x-axis to the upper curve, and the inner radius is the distance from the x-axis to the lower curve within the region. Since \( y = 0 \) is one boundary, the radius will be \( y \) itself.
Determine the limits of integration for \( y \) by finding the intersection points of the curves in terms of \( y \). Solve for the values of \( y \) where the curves intersect to establish the bounds of integration.
Set up the integral for the volume using the washer method formula: \[ V = \pi \int_{a}^{b} \left( R_{outer}(y)^2 - R_{inner}(y)^2 \right) \, dy \], where \( R_{outer}(y) \) and \( R_{inner}(y) \) are the outer and inner radii of the washers. Express these radii in terms of \( y \) using the given curves.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Region Bounded by Curves
Understanding the region R requires identifying the area enclosed by the given curves. This involves finding the points of intersection and determining which curve lies above or below within the interval. Properly sketching or visualizing the region helps set up integrals accurately.
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Finding Area When Bounds Are Not Given
Volume of Solids of Revolution
When a region is revolved around an axis, it generates a 3D solid. The volume can be found using methods like the disk/washer or shell method, which involve integrating cross-sectional areas perpendicular to the axis of rotation.
Recommended video:
04:48Finding Volume Using Disks
Setting up a Single Integral for Volume
To write a single integral for volume, one must express the radius and limits of integration in terms of a single variable. This often requires rewriting curves and choosing the appropriate method (disk/washer or shell) to represent the volume as an integral with clear bounds.
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05:38Introduction to Cross Sections
Related Practice
Textbook Question
27–33. Multiple regions The regions R₁,R₂, and R₃ (see figure) are formed by the graphs of y = 2√x,y = 3−x,and x=3.
Find the area of each of the regions R₁,R₂, and R₃.
Textbook Question
58–61. Arc length Find the length of the following curves.
y = 2x+4 on [−2,2] (Use calculus.)
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Textbook Question
43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.
The region bounded by the curve y = 1+√x, the curve y = 1−√x, and the line x=1 is revolved about the y-axis. Find the volume of the resulting solid by (a) integrating with respect to x and (b) integrating with respect to y. Be sure your answers agree.
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Textbook Question
43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.
The region bounded by the graphs of y = 2x,y = 6−x, and y = 0 is revolved about the line y = −2 and the line x = −2. Find the volumes of the resulting solids. Which one is greater?
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Textbook Question
27–33. Multiple regions The regions R₁,R₂, and R₃ (see figure) are formed by the graphs of y = 2√x,y = 3−x,and x=3.
Use the shell method to find an integral, or sum of integrals, that equals the volume of the solid obtained by revolving region R₃ about the line x=3. Do not evaluate the integral.
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