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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 6.4.37
Chapter 6, Problem 6.4.37

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 = (x−2)³ −2,x=0, and y=25; about the y-axis

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First, identify the region R bounded by the curves: \(y = (x - 2)^3 - 2\), the vertical line \(x = 0\), and the horizontal line \(y = 25\). Sketching or visualizing this region helps understand the limits of integration and the shape of the solid.
For the shell method (revolving around the y-axis), express the radius and height of a typical cylindrical shell. The radius is the distance from the y-axis, which is \(x\), and the height is the vertical distance between the curves, which can be written in terms of \(x\) as \(y_{top} - y_{bottom}\). Here, the height is \(25 - ((x - 2)^3 - 2)\).
Set up the integral for the shell method. The volume \(V\) is given by \(V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dx\). Determine the limits of integration \(a\) and \(b\) by finding the \(x\)-values where the region is bounded, specifically between \(x=0\) and the \(x\)-value where \(y=25\) intersects \(y = (x-2)^3 - 2\).
For the washer method (also revolving around the y-axis), express \(x\) as a function of \(y\) by solving \(y = (x - 2)^3 - 2\) for \(x\). This gives \(x = 2 + \sqrt[3]{y + 2}\). The outer radius is the distance from the y-axis to the outer curve, and the inner radius is the distance to the inner curve (here, \(x=0\)).
Set up the integral for the washer method. The volume \(V\) is given by \(V = \pi \int_{c}^{d} \left[(\text{outer radius})^2 - (\text{inner radius})^2\right] dy\). The limits \(c\) and \(d\) correspond to the \(y\)-values bounding the region, from \(y = -2\) (when \(x=0\)) up to \(y=25\). Write the integral explicitly using these limits and the expressions for the radii.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Shell Method

The shell method calculates the volume of a solid of revolution by summing cylindrical shells. Each shell's volume is found by multiplying its circumference, height, and thickness. This method is especially useful when revolving around an axis parallel to the axis of the function.
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Euler's Method

Washer Method

The washer method finds volume by slicing the solid perpendicular to the axis of revolution, creating washers (disks with holes). The volume is the integral of the difference between the outer and inner radii squared, times π, over the interval. It works well when the solid has a hollow center.
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Euler's Method

Setting up the Region and Limits of Integration

Identifying the region bounded by the curves and determining the correct limits of integration is crucial. This involves solving for intersection points and expressing variables appropriately for integration with respect to x or y, depending on the method and axis of revolution.
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Integration by Parts for Definite Integrals Example 8
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