Textbook Question
21–30. {Use of Tech} Arc length by calculator
a. Write and simplify the integral that gives the arc length of the following curves on the given interval.
y = 1/x, for 1 ≤ x ≤ 10
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 6.3.10a
Region R is revolved about the line x=4 to form a solid of revolution.
a. What is the radius of a cross section of the solid at a point y in [1, 3]?
Verified step by step guidance1
Step 1: Understand the problem setup. The region R is revolved about the vertical line x=4, forming a solid of revolution. The radius of a cross-section at a given y-value is the horizontal distance between the line x=4 and the x-coordinate of the region R at that y-value.
Step 2: Identify the x-coordinate of the region R at a given y-value. If the region R is defined by a function x=f(y), then the x-coordinate at a specific y-value is given by f(y).
Step 3: Calculate the radius of the cross-section. The radius is the absolute difference between the line x=4 and the x-coordinate of the region R. Mathematically, this is expressed as: .
Step 4: Ensure the y-value is within the given interval [1, 3]. The radius calculation is valid only for y-values in this range, as specified in the problem.
Step 5: Substitute the specific function f(y) (if provided) into the formula for the radius to express the radius explicitly in terms of y. If no function is given, the radius remains in the general form: .
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Key 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 a straight line (axis of rotation). The shape's volume can be calculated using methods such as the disk or washer method, depending on the orientation of the axis and the region being revolved.
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04:48
Finding Volume Using Disks
Radius of Cross Section
The radius of a cross section of a solid of revolution is the distance from the axis of rotation to the edge of the region being revolved. This radius varies depending on the specific point along the axis and is crucial for calculating the volume of the solid using integration.
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06:18Introduction to Cross Sections Example 2
Integration in Volume Calculation
Integration is a fundamental concept in calculus used to find the volume of solids of revolution. By integrating the area of cross sections along the axis of rotation, one can determine the total volume of the solid. The specific method of integration (disk, washer, or cylindrical shells) depends on the geometry of the solid and the axis of rotation.
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04:48Finding Volume Using Disks
Related Practice
Textbook Question
55–58. Marginal cost Consider the following marginal cost functions.
a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.
C′(x)=200−0.05x
Textbook Question
Emptying a water trough A water trough has a semicircular cross section with a radius of 0.25 m and a length of 3 m (see figure).
a. How much work is required to pump the water out of the trough (to the level of the top of the trough) when it is full?
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Textbook Question
A vertical spring A 10-kg mass is attached to a spring that hangs vertically and is stretched 2 m from the equilibrium position of the spring. Assume a linear spring with F(x) = kx.
a. How much work is required to compress the spring and lift the mass 0.5 m?
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Textbook Question
17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.
a. Determine the position function, for t≥0, using the antiderivative method
v(t) = 6−2t on [0, 5]; s(0)=0
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Textbook Question
A right circular cylinder with height R and radius R has a volume of VC=πR^3 (height = radius).
a. Find the volume of the cone that is inscribed in the cylinder with the same base as the cylinder and height R. Express the volume in terms of VC.
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