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) = 300+10x−0.01x²
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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 6.4.8a
6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.
Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.
a. What is the radius of a cylindrical shell at a point x in [0, 4]?
Verified step by step guidance1
Identify the axis of rotation: The region R is revolved about the vertical line x=4. This means the radius of a cylindrical shell at a point x is the horizontal distance from x to the line x=4.
Determine the formula for the radius: The radius is given by the difference between the x-coordinate of the line of rotation (x=4) and the x-coordinate of the shell. Thus, the radius is r(x) = 4 - x.
Verify the bounds of integration: The region R is bounded by x-values from x=0 to x=4, so the radius formula r(x) = 4 - x is valid for all x in [0, 4].
Understand the shell method setup: In the shell method, the radius r(x) is multiplied by the height of the shell and integrated over the bounds of x to compute the volume. For this part of the problem, we are only determining the radius.
Conclude the radius: The radius of a cylindrical shell at a point x in [0, 4] is r(x) = 4 - x.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cylindrical Shell Method
The cylindrical shell method is a technique for finding the volume of a solid of revolution. It involves slicing the solid into thin cylindrical shells, where the volume of each shell is calculated and then integrated over the interval of interest. This method is particularly useful when revolving a region around a vertical line, as it simplifies the calculation of the radius and height of each shell.
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Euler's Method
Radius of a Shell
In the context of the shell method, the radius of a cylindrical shell is the distance from the axis of rotation to the shell itself. For the given problem, where the region R is revolved around the line x=4, the radius at a point x is determined by the difference between the line x=4 and the x-coordinate of the shell, which is expressed as (4 - x). This distance is crucial for calculating the volume of the shell.
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Bounded Region
The bounded region R in the problem is defined by the curves y = 2, y = 2 - √x, and the vertical line x = 4. Understanding the boundaries of this region is essential for setting up the integral for volume calculation. The area enclosed by these curves in the first quadrant provides the shape that will be revolved around the line x=4 to generate the solid.
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07:45Area of Polar Regions
Related Practice
Textbook Question
Given the velocity function of an object moving along a line, explain how definite integrals can be used to find the displacement of the object.
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Textbook Question
{Use of Tech} Oscillating motion A mass hanging from a spring is set in motion, and its ensuing velocity is given by v(t) = 2π cos πt, for t≥0. Assume the positive direction is upward and s(0)=0.
a. Determine the position function, for t≥0.
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Textbook Question
Depletion of natural resources Suppose r(t) = r0e^−kt, with k>0, is the rate at which a nation extracts oil, where r0=10⁷ barrels/yr is the current rate of extraction. Suppose also that the estimate of the total oil reserve is 2×10⁹ barrels.
a. Find Q(t), the total amount of oil extracted by the nation after t years.
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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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