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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.AAE.18a
Chapter 8, Problem 8.AAE.18a

18. Finding volume (Continuation of Exercise 17.) Find the volume of the solid generated by revolving the region R about:
a. the y-axis.

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1
Identify the region R from Exercise 17. Typically, this region is bounded by given curves or lines. Make sure you clearly understand the boundaries of R before proceeding.
Since the solid is generated by revolving the region R about the y-axis, decide on the method to use for finding the volume. The common methods are the Disk/Washer method or the Shell method. For revolution around the y-axis, the Shell method is often convenient.
Set up the integral using the Shell method. The formula for the volume using cylindrical shells when revolving around the y-axis is: \[ V = \int_{a}^{b} 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx \] Here, the radius is the distance from the y-axis (which is simply \(x\)), and the height is the vertical length of the region at each \(x\).
Determine the limits of integration \(a\) and \(b\) from the domain of the region R along the x-axis. Also, express the height of the shell as a function of \(x\) by subtracting the lower curve from the upper curve within the region.
Write the integral explicitly by substituting the radius and height expressions, then prepare to evaluate the integral to find the volume. Remember, do not compute the final value yet—just set up the integral correctly.

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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 calculating the volume of a 3D solid formed by rotating a 2D region around an axis. The volume is typically found using integral calculus methods such as the disk, washer, or shell methods, depending on the axis of rotation and the shape of the region.
Recommended video:
04:48
Finding Volume Using Disks

Shell Method

The shell method calculates volume by integrating cylindrical shells formed by revolving vertical or horizontal slices of the region around an axis. It is especially useful when revolving around the y-axis and when the region is described in terms of x.
Recommended video:
07:33
Euler's Method

Axis of Rotation (y-axis)

The axis of rotation determines how the region is revolved to form the solid. Revolving around the y-axis means the solid is generated by rotating the region horizontally, which affects the choice of method and the limits of integration.
Recommended video:
06:30
Disk Method Using y-Axis
Related Practice
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The region in the first quadrant enclosed by the coordinate axes, the curve y = e^x, and the line x = 1 is revolved about the y-axis to generate a solid. Find the volume of the solid.

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Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

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