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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.4.8c
Chapter 6, Problem 6.4.8c

6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.
Graph showing region R bounded by y=2, y=2−√x, and x=4 in the first quadrant, illustrating the shell method for volume.
Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.
c. Write an integral for the volume of the solid using the shell method.

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Step 1: Understand the shell method. The shell method involves integrating the volume of cylindrical shells formed by revolving a region around a vertical or horizontal axis. The formula for the volume is: \( V = \int_{a}^{b} 2\pi \cdot \text{radius} \cdot \text{height} \cdot dx \).
Step 2: Identify the radius and height of the shells. Since the region R is revolved around the line \( x = 4 \), the radius of a shell is \( 4 - x \) (distance from the line \( x = 4 \) to a point \( x \) in the region). The height of the shell is given by the difference between the upper curve \( y = 2 \) and the lower curve \( y = 2 - \sqrt{x} \).
Step 3: Set up the integral bounds. The region R is bounded horizontally between \( x = 0 \) and \( x = 4 \). These will be the limits of integration.
Step 4: Write the integral expression for the volume. Using the shell method formula, the integral becomes: \( V = \int_{0}^{4} 2\pi (4 - x) \cdot [(2) - (2 - \sqrt{x})] \cdot dx \).
Step 5: Simplify the height expression inside the integral. The height \( (2) - (2 - \sqrt{x}) \) simplifies to \( \sqrt{x} \). Thus, the integral for the volume becomes: \( V = \int_{0}^{4} 2\pi (4 - x) \cdot \sqrt{x} \cdot dx \).

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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

In this problem, the region R is defined by the curves y = 2, y = 2 - √x, and the vertical line x = 4. Understanding how to identify and sketch the area bounded by these curves is crucial for visualizing the solid of revolution. The intersection points of these curves help determine the limits of integration when calculating volume.
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Shell Method

The shell method is a technique for finding the volume of a solid of revolution. It involves slicing the solid into cylindrical shells and integrating the volume of these shells. When revolving around a vertical line, the formula for the volume V is given by V = 2π ∫ (radius)(height) dx, where the radius is the distance from the axis of rotation to the shell and height is the function value.
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Setting Up the Integral

To write the integral for the volume using the shell method, one must determine the appropriate limits of integration and the expressions for the radius and height. In this case, the radius is (4 - x) and the height is (2 - (2 - √x)) = √x. The integral will be set up from x = 0 to x = 4, capturing the entire region R as it revolves around the line x = 4.
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Related Practice
Textbook Question

Piecewise velocity The velocity of a (fast) automobile on a straight highway is given by the function

v(t)={3t if 0≤t<2060 if 20≤t<45240−4t if t≥45 v(t)= \(\begin{cases}\)3 t & \(\text\) { if } 0 \(\leq\) t<20 \\ 60 & \(\text\) { if } 20 \(\leq\) t<45 \\ 240-4 t & \(\text\) { if } t \(\geq\) 45\(\end{cases}\)

, where is measured in seconds and v has units of m/s. 


c. What is the distance traveled by the automobile in the first 60 s?

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample. 


c. Let f(x)=12x^2. The area of the surface generated when the graph of f on [−4, 4] is revolved about the x-axis is twice the area of the surface generated when the graph of f on [0, 4] is revolved about the x-axis. 

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Textbook Question

Let R be the region in the first quadrant bounded above by the curve y=2−x² and bounded below by the line y=x. Suppose the shell method is used to determine the volume of the solid generated by revolving R about the y-axis.

c. Write an integral for the volume of the solid using the shell method.

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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. 


c. At what times does the mass reach its low point the first three times? 

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


Arc length may be negative if f(x) < 0 on part of the interval in question.

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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.


c. What is the area A(y) of a cross section of the solid at a point y in [1, 3]?

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