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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 6.4.5a
Chapter 6, Problem 6.4.5a

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 showing the shell method for volume calculation, with curves and cylindrical shells in the first quadrant.
a. What is the radius of a cylindrical shell at a point x in [0, 2]?

Verified step by step guidance
1
Identify the axis of rotation, which is the y-axis in this problem.
Recall that when using the shell method to revolve a region around the y-axis, the radius of a cylindrical shell at a point x is the horizontal distance from the y-axis to the shell.
Since the y-axis is at x = 0, the radius of the shell at a point x is simply the x-coordinate itself.
Therefore, the radius of a cylindrical shell at a point x in the interval [0, 2] is given by the expression \(x\).
This radius represents the distance from the y-axis to the vertical shell located at x, which will be used in the volume integral.

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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 Calculation

The shell method calculates the volume of a solid of revolution by summing cylindrical shells formed by revolving vertical slices of a region around an axis. Each shell's volume is approximated by 2π(radius)(height)(thickness), where the radius is the distance from the axis of rotation to the shell, height is the function value difference, and thickness is a small change in x.
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Finding Volume Using Disks

Radius of a Cylindrical Shell

In the shell method, the radius of a shell is the distance from the axis of rotation to the vertical slice at a given x-value. When revolving around the y-axis, the radius is simply the x-coordinate of the shell, since the shell is at position x units from the y-axis.
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Radius of Convergence

Region Bounded by Curves

The region R is bounded above by y = 2 - x² and below by y = x in the first quadrant. Understanding these boundaries helps determine the height of each shell as the vertical distance between the two curves at a given x, which is essential for setting up the integral for volume.
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Related Practice
Textbook Question

13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.


a. Determine when the motion is in the positive direction and when it is in the negative direction. 


v(t) = 50e^−2t on [0, 4]

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Textbook Question

21–30. {Use of Tech} Arc length by calculator


a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 

y = 1/x, for 1 ≤ x ≤ 10

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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)=200−0.05x

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Textbook Question

Emptying a cylindrical tank A cylindrical water tank has height 8 m and radius 2m (see figure).

a. If the tank is full of water, how much work is required to pump the water to the level of the top of the tank and out of the tank?

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Textbook Question

Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions.

a. ∫a^b √1+16x⁴ dx

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Textbook Question

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


a. The distance traveled by an object moving along a line is the same as the displacement of the object.

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