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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 6.4.30
Chapter 6, Problem 6.4.30

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 = In x/x²,y = 0,x = 3, about the y-axis

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First, identify the region R bounded by the curves: \(y = \frac{\ln x}{x^2}\), \(y = 0\), and \(x = 3\). Since the region is revolved about the y-axis, we will use the shell method with respect to \(x\).
Recall the shell method formula for volume when revolving around the y-axis: \(V = 2\pi \int_a^b (\text{radius})(\text{height}) \, dx\). Here, the radius of a shell is the distance from the y-axis, which is \(x\), and the height is the function value \(y = \frac{\ln x}{x^2}\).
Set up the integral limits from \(x = 1\) to \(x = 3\) because \(y = \frac{\ln x}{x^2}\) is defined and positive between these points, and the region is bounded by \(y=0\) (the x-axis). Note that \(x=1\) is where \(y=0\) since \(\ln 1 = 0\).
Write the volume integral as: \(V = 2\pi \int_1^3 x \cdot \frac{\ln x}{x^2} \, dx\). Simplify the integrand to \(2\pi \int_1^3 \frac{\ln x}{x} \, dx\).
To find the volume, evaluate the integral \(\int_1^3 \frac{\ln x}{x} \, dx\) using integration techniques such as substitution or integration by parts, then multiply the result by \(2\pi\).

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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 approximated by its circumference times height times thickness. When revolving around the y-axis, shells are vertical slices parallel to the axis, and the radius is the x-value of the shell.
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Setting up the Integral with Given Curves

To use the shell method, identify the height and radius of each shell from the given curves. Here, the height is the function y = (ln x) / x², bounded below by y = 0, and the radius is the distance from the y-axis, which is x. The limits of integration are from x = 1 (where ln x / x² > 0) to x = 3.
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Properties of the Function y = (ln x) / x²

Understanding the behavior of y = (ln x) / x² is crucial for setting correct bounds and ensuring the function is positive over the interval. The natural logarithm ln x is positive for x > 1, and dividing by x² affects the shape, so the region lies above y=0 between x=1 and x=3.
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Related Practice
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