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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 6.4.25
Chapter 6, Problem 6.4.25

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. 


{Use of Tech} y = 1 / (x² + 1)²,y=0,x=1, and x=2; about the y-axis

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Identify the region R bounded by the curves: \( y = \frac{1}{(x^2 + 1)^2} \), \( y = 0 \), \( x = 1 \), and \( x = 2 \). This region lies between \( x = 1 \) and \( x = 2 \) above the x-axis and under the curve \( y = \frac{1}{(x^2 + 1)^2} \).
Since the solid is generated by revolving the region about the y-axis, use the shell method. The shell method formula for volume when revolving around the y-axis is: \[ V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dx \] where the radius is the distance from the y-axis to the shell (which is \( x \)) and the height is the function value \( y = \frac{1}{(x^2 + 1)^2} \).
Set up the integral with the limits of integration from \( x = 1 \) to \( x = 2 \): \[ V = 2\pi \int_{1}^{2} x \cdot \frac{1}{(x^2 + 1)^2} \, dx \]
Simplify the integrand if possible. Here, the integrand is \( \frac{x}{(x^2 + 1)^2} \). Consider using substitution to evaluate the integral later, such as letting \( u = x^2 + 1 \), which implies \( du = 2x \, dx \).
After setting up the integral, proceed to evaluate it using the substitution method or other integration techniques. Finally, multiply the result by \( 2\pi \) to find the volume of the solid.

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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 revolving a region around an axis parallel to the axis of the variable of integration.
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Setting up the Integral with Given Bounds

To apply the shell method, identify the radius and height of each shell based on the region's boundaries. Here, the radius is the distance from the y-axis (the axis of rotation) to x, and the height is given by the function y = 1/(x² + 1)². The bounds for x are from 1 to 2, defining the limits of integration.
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Understanding the Function and Region

The function y = 1/(x² + 1)² describes the upper boundary of the region, while y = 0, x = 1, and x = 2 form the other boundaries. Recognizing these curves helps visualize the region being revolved and ensures correct interpretation of the height and limits in the integral.
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Related Practice
Textbook Question

A solid has a circular base; cross sections perpendicular to the base are squares. What method should be used to find the volume of the solid?

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Force on the end of a tank Determine the force on a circular end of the tank in Figure 6.78 if the tank is full of gasoline. The density of gasoline is ρ = 737 kg/m³.

Textbook Question

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.

y = x,y = 2x+2,x = 2, and x=6; about the y-axis

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=x,y=x+2,x=0, and x=4 ; about the x-axis

Textbook Question

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


The region bounded by y=4x+4, y=6x+6, and x=4

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Textbook Question

Find the arc length of the line y = 4−3x on [−3, 2] using calculus and verify your answer using geometry.