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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 6.4.22
Chapter 6, Problem 6.4.22

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. 


x = x³ ,y = 1, and x = 0; about the x-axis

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1
First, identify the region R bounded by the curves: the curve given by \(y = x^3\), the line \(y = 1\), and the vertical line \(x = 0\). Sketching this region helps visualize the problem.
Since the solid is generated by revolving the region about the x-axis, and the shell method is requested, we consider cylindrical shells formed by slicing the region parallel to the axis of revolution. Here, it is easier to use horizontal slices (in terms of \(y\)) because the shells will be vertical.
Express \(x\) as a function of \(y\) from the curve \(y = x^3\). Solving for \(x\) gives \(x = y^{1/3}\). This will help determine the radius and height of each shell.
Set up the shell method integral. The radius of a shell is the distance from the shell to the axis of rotation (the x-axis), which is \(y\). The height of the shell is the horizontal length of the region at height \(y\), which is from \(x=0\) to \(x = y^{1/3}\), so the height is \(y^{1/3} - 0 = y^{1/3}\).
The volume integral using the shell method is then \(V = 2\pi \int_{y=0}^{y=1} (\text{radius})(\text{height}) \, dy = 2\pi \int_0^1 y \cdot y^{1/3} \, dy\). Simplify the integrand and prepare to integrate over \(y\) from 0 to 1.

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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 integrating cylindrical shells. Each shell's volume is found by multiplying its circumference, height, and thickness. This method is especially useful when the axis of rotation is parallel to the axis of the variable of integration.
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Finding Volume Using Disks

Setting up the Region and Bounds

Identifying the region bounded by the given curves is crucial. Here, the region is bounded by y = x³, y = 1, and x = 0. Understanding these boundaries helps determine the limits of integration and the dimensions of each shell when revolving around the x-axis.
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Area of Polar Regions

Revolution about the x-axis

Revolving a region about the x-axis means the shells are formed vertically with respect to y. The radius of each shell corresponds to the distance from the x-axis, and the height is determined by the horizontal length of the region at that y-value. This affects how the integral is set up.
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Disk Method Using y-Axis
Related Practice
Textbook Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.


y=4−x^2,x=2, and y=4; about the y-axis

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Textbook Question

29–36. Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2).


a(t) = −32; v(0)=50; s(0)=0

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Textbook Question

Express the area of the shaded region in Exercise 5 as the sum of two integrals with respect to y. Do not evaluate the integrals.

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Textbook Question

Look again at the region R in Figure 6.38 (p. 439). Explain why it would be difficult to use the washer method to find the volume of the solid of revolution that results when R is revolved about the y-axis.

53
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Textbook Question

35–38. Shell and washer methods Let R be the region bounded by the following curves. Use both the shell method and the washer method to find the volume of the solid generated when R is revolved about the indicated axis.


y = 8,y = 2x+2,x = 0, and x=2; about the y-axis

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Textbook Question

Find the area of the region described in the following exercises.


The region bounded by y=√x, y=2x−15, and y=0

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