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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 6.4.5c
Chapter 6, Problem 6.4.5c

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.
Illustration of the shell method for calculating volume, showing region R, shell height, and radius in the first quadrant.
c. Write an integral for the volume of the solid using the shell method.

Verified step by step guidance
1
Identify the region R bounded by the curves y = 2 - x^2 (above) and y = x (below) in the first quadrant. This region is revolved around the y-axis to form the solid.
Recall that the shell method involves integrating with respect to x when revolving around the y-axis. Each shell has a radius equal to the distance from the y-axis, which is x, and a height equal to the difference between the upper and lower functions: height = (2 - x^2) - x.
Write the volume of a typical shell as the circumference times the height times the thickness: Volume of shell = 2\(\pi\) \(\times\) (radius) \(\times\) (height) \(\times\) (thickness) = 2\(\pi\) x \(\big\)((2 - x^2) - x\(\big\)) \, dx.
Determine the limits of integration by finding the x-values where the two curves intersect in the first quadrant. Solve 2 - x^2 = x to find these points.
Set up the integral for the volume as: \(V = \int_{a}^{b} 2\pi x \big((2 - x^2) - x\big) \, dx\), where a and b are the intersection points found in the previous step.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Shell Method for Volume

The shell method calculates the volume of a solid of revolution by summing cylindrical shells. Each shell's volume is approximated by 2π(radius)(height)(thickness), where the radius is the distance from the axis of rotation, the height is the function value difference, and the thickness is a small change in the variable of integration.
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Setting up the Integral with Respect to x

When revolving around the y-axis, the shell radius is the x-value of the shell, and the height is the difference between the upper and lower functions, here y=2−x² and y=x. The integral sums shells from the leftmost to rightmost x-values defining the region, integrating with respect to x.
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Determining the Bounds of Integration

The bounds are found by identifying where the curves intersect in the first quadrant. Solving y=2−x² and y=x gives the limits for x, which define the interval over which the shells are integrated to find the volume.
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Related Practice
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

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.


c. If the probe was released from an altitude of 3 km, when does it enter the ocean?

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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

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

Flow rates in the Spokane River The daily discharge of the Spokane River as it flows through Spokane, Washington, in April and June is modeled by the functions

r1(t) = 0.25t²+37.46t+722.47 (April) and

r2(t) = 0.90t²−69.06t+2053.12 (June), where the discharge is measured in millions of cubic feet per day, and t=0 corresponds to the beginning of the first day of the month (see figure).

c. The Spokane River flows out of Lake Coeur d’Alene, which contains approximately 0.67mi³ of water. Determine the percentage of Lake Coeur d’Alene’s volume that flows through Spokane in April and June.

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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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