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5.3 Volumes of Revolution: Shell Method

The Shell Method is a technique for calculating volumes of solids of revolution, placing the typical element parallel to the axis of rotation. Unlike the Disk Method, which positions the element perpendicularly, the Shell Method generates a cylindrical shell. The volume calculation involves determining the integration variable based on the width of the element—dx for horizontal elements and dy for vertical elements. This guide reviews key concepts and provides exercises to practice applying the Shell Method for volume generation by revolving regions about the x-axis and y-axis.

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5.3 Volumes of Revolution: Shell Method

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  1. 5.3 Volumes of Revolution: Shell Method Remember, in the Disk Method we place the typical element perpendicular to the axis of rotation. Now, for the Shell Method, we place the typical element parallel to the axis of rotation. When we rotate it, it produces a cylindrical shell. But, in both methods, if the width of the element is dx, then the variable of integration is x; if the width of the element is dy, then the variable of integration is y. Important: for the Disk Method, the elementary volume was For the Shell Method, the elementary volume is 1

  2. Exercise: Use the shell method to find the volume generated by revolving the region bounded by a. about the x-axis b. about the y-axis 2

  3. Homework: Section 5.3: 3,15,19,27,41. 3

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