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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 6.3.8c
Chapter 6, Problem 6.3.8c

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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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 y-axis, the resulting solid has cross sections that are washers. We need to find the area A(y) of a cross section at a point y in the interval [1, 3].
Step 2: Express x in terms of y using the equation y = 1 + √x. Rearrange this equation to isolate x: √x = y - 1, so x = (y - 1)^2.
Step 3: Identify the inner and outer radii of the washer. The outer radius is determined by the curve x = (y - 1)^2, and the inner radius is determined by the vertical line x = 4. Since the solid is revolved about the y-axis, the radii are measured horizontally from the y-axis.
Step 4: Write the formula for the area of a washer. The area A(y) of a cross section is given by the formula: A(y) = π[R_outer^2 - R_inner^2], where R_outer is the distance from the y-axis to the curve x = (y - 1)^2, and R_inner is the distance from the y-axis to the line x = 4.
Step 5: Substitute the expressions for R_outer and R_inner into the formula. R_outer = (y - 1)^2 and R_inner = 4. Therefore, A(y) = π[((y - 1)^2)^2 - 4^2]. Simplify this expression to find the area A(y) in terms of y.

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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 an axis. In this case, the region R is revolved around the y-axis, resulting in a solid whose volume can be calculated using methods such as the disk or washer method. Understanding this concept is crucial for visualizing the shape formed and for applying the appropriate formulas to find its volume.
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Finding Volume Using Disks

Washer Method

The washer method is a technique used to calculate the volume of a solid of revolution when the cross-sections are washers, which are circular disks with a hole in the center. The volume is determined by integrating the area of these washers along the axis of rotation. For the given problem, identifying the outer and inner radii of the washers at a specific y-value is essential for finding the area A(y) of the cross-section.
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Euler's Method

Area of Cross Section

The area of a cross section refers to the area of the shape formed when a solid is sliced perpendicular to an axis. In this context, A(y) represents the area of the washer at a specific height y. To find A(y), one must calculate the difference between the areas of the outer and inner circles defined by the functions bounding the region, which is critical for determining the volume of the solid formed by revolving the region.
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Introduction to Cross Sections
Related Practice
Textbook Question

A nonlinear spring Hooke’s law is applicable to idealized (linear) springs that are not stretched or compressed too far from their equilibrium positions. Consider a nonlinear spring whose restoring force is given by F(x) = 16x−0.1x³, for |x|≤7. 

c. How much work is done in compressing the spring from its equilibrium position (x=0) to x=−2?

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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≥45v(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

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

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.

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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