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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 6.4.57
Chapter 6, Problem 6.4.57

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

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First, identify the region R bounded by the curves: \(y = x\), \(y = 2x + 2\), and the vertical lines \(x = 2\) and \(x = 6\). Sketching these curves and lines will help visualize the region and the axis of rotation (the y-axis).
Since the solid is generated by revolving the region around the y-axis, decide on the method to use. Here, the shell method is often convenient because the axis of rotation is vertical and the region is bounded by vertical lines.
Set up the shell method integral. The formula for the volume using cylindrical shells when revolving around the y-axis is: \(V = \int_{a}^{b} 2\pi (\text{radius})(\text{height}) \, dx\) where the radius is the distance from the y-axis (which is \(x\)), and the height is the difference between the top and bottom functions in terms of \(y\) for each \(x\).
Determine the height of each shell by finding the difference between the two curves in terms of \(y\): Height \(= (2x + 2) - x = x + 2\) The radius is simply \(x\), and the limits of integration are from \(x=2\) to \(x=6\).
Write the integral for the volume: \(V = \int_{2}^{6} 2\pi \cdot x \cdot (x + 2) \, dx\) This integral can then be evaluated 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.

Volume of Solids of Revolution

This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. The volume can be computed using integral calculus by summing infinitesimal cross-sectional areas perpendicular to the axis of rotation.
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Finding Volume Using Disks

Methods of Integration: Disk, Washer, and Shell

These are techniques to calculate volumes of solids of revolution. The disk and washer methods integrate cross-sectional areas perpendicular to the axis, while the shell method integrates cylindrical shells parallel to the axis, often simplifying calculations depending on the axis and region.
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Disk Method Using y-Axis

Setting up the Integral with Given Boundaries

Accurately identifying the limits of integration and expressing the radius and height of slices or shells in terms of the variable of integration is crucial. This involves understanding the bounding curves and the axis of rotation to correctly formulate the integral.
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Integration by Parts for Definite Integrals Example 8
Related Practice
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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=2x, and y=6 ; about the y-axis

Textbook Question

Use the general slicing method to find the volume of the following solids.

The solid whose base is the region bounded by the semicircle y=√1−x^2 and the x-axis, and whose cross sections through the solid perpendicular to the x-axis are squares

Textbook Question

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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