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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

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

Bike race Theo and Sasha start at the same place on a straight road, riding bikes with the following velocities (measured in mi/hr). Assume t is measured in hours.

Theo: vT(t)=10, for t≥0

Sasha: vS(t)=15t, for 0≤t≤1, and vS(t)=15, for t>1


c. If the riders ride for 2 hr, who rides farther? Interpret your answer geometrically using the graphs of part (a). 

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Textbook Question

9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

d. A piecewise function for s(t)

Textbook Question

Filling a tank A 2000-liter cistern is empty when water begins flowing into it (at t=0 at a rate (in L/min) given by Q′(t) = 3√t, where t is measured in minutes.


c. When will the tank be full?

Textbook Question

Let R be the region bounded by the curve y=√cos x and the x-axis on [0, π/2]. A solid of revolution is obtained by revolving R about the x-axis (see figures). 


c. Write an integral for the volume of the solid.

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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.