Textbook Question
Work done by a spring A spring on a horizontal surface can be stretched and held 0.5 m from its equilibrium position with a force of 50 N.
b. How much work is done in compressing the spring 0.5 m from its equilibrium position?
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 6.3.7b
Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.
Region R is revolved about the x-axis to form a solid of revolution whose cross sections are washers.
b. What is the inner radius of a cross section of the solid at a point x in [0, 4]?
Verified step by step guidance1
Step 1: Understand the problem. The region R is bounded by the graphs of y = 1 + √x, x = 4, and y = 1. When this region is revolved about the x-axis, the resulting solid has cross sections that are washers. The goal is to determine the inner radius of a washer at a point x in the interval [0, 4].
Step 2: Recall the formula for the inner radius of a washer. The inner radius is the distance from the x-axis to the inner curve of the region being revolved. In this case, the inner curve is the horizontal line y = 1.
Step 3: Analyze the geometry of the region. At any point x in [0, 4], the inner radius is determined by the vertical distance from the x-axis to the line y = 1. Since the line y = 1 is constant, the inner radius does not depend on x.
Step 4: Express the inner radius mathematically. The inner radius is simply the value of y for the inner curve, which is y = 1. Therefore, the inner radius is 1 for all x in the interval [0, 4].
Step 5: Conclude that the inner radius of the washer at any point x in [0, 4] is constant and equal to 1. This is because the inner curve is a horizontal line at y = 1.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Solid of Revolution
A solid of revolution is a three-dimensional shape created by rotating a two-dimensional area around an axis. In this case, the region R is revolved around the x-axis, forming a solid. Understanding this concept is crucial for visualizing the shape and determining its volume using integration techniques.
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Finding Volume Using Disks
Washer Method
The washer method is a technique used to find the volume of a solid of revolution when the cross-sections are washers. It involves subtracting the volume of the inner solid from the outer solid. The inner radius and outer radius are essential for setting up the integral that calculates the volume of the solid formed by revolving the region around the axis.
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Inner Radius
The inner radius in the context of the washer method refers to the distance from the axis of rotation to the inner curve of the region being revolved. For the given problem, it is the distance from the x-axis to the curve defined by y=1. Identifying this radius is necessary to accurately compute the volume of the solid formed by the revolution.
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Related Practice
Textbook Question
In the design of solid objects (both artificial and natural), the ratio of the surface area to the volume of the object is important. Animals typically generate heat at a rate proportional to their volume and lose heat at a rate proportional to their surface area. Therefore, animals with a low SAV ratio tend to retain heat, whereas animals with a high SAV ratio (such as children and hummingbirds) lose heat relatively quickly.
b. What is the SAV ratio of a ball with radius a?
Textbook Question
Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.
Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers.
b. What is the inner radius of a cross section of the solid at a point y in [1, 3]?
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Textbook Question
Region R is revolved about the line y=1 to form a solid of revolution.
c. Write an integral for the volume of the solid.
Textbook Question
Probe speed A data collection probe is dropped from a stationary balloon, and it falls with a velocity (in m/s) given by v(t) = 9.8t, neglecting air resistance. After 10 s, a chute deploys and the probe immediately slows to a constant speed of 10 m/s, which it maintains until it enters the ocean.
b. How far does the probe fall in the first 30 s after it is released?
Textbook Question
Let R be the region bounded by the curve y=cos^−1x and the x-axis on [0, 1]. A solid of revolution is obtained by revolving R about the y-axis (see figures).
b. Find an expression for the area A(y) of a cross section of the solid at a point y in [0,π/2].