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Calculating Volume of Revolved Solids Using Disk Method

This guide walks you through finding the volume of solids formed by revolving regions around axes using the disk method. We start by determining the area of region R in terms of x, set up the appropriate integral, and compute the volume by integrating the area of the disks. For a region R bounded by x = 0 to x = 3, we find the area of cross-sections that are disks. Additionally, we explore the volume of solids formed by revolving regions about the y-axis and provide detailed steps for integration with respect to both x and y for a comprehensive understanding.

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Calculating Volume of Revolved Solids Using Disk Method

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Presentation Transcript

  1. Aim: How do we find the volume by disk method? Do Now: Find the area of R 2. Set up integral 3. compute 1. dy or dx ? R 3

  2. Find the volume of solid formed by revolving R about x-axis Find the area of the cross-section that is a disk R r 3 To find the volume, we to integrate the area of the disk from x = 0 to x = 3

  3. Find the area of S y 9 It’s easier to integrate in terms of y S x 3

  4. Find the volume of the solid formed by revolving S about the y-axis y 9 S r = x r x 3

  5. Find the volume of the solid by revolving R about x = 3 R dy r 3 )

  6. Find the volume of the solid formed by revolving S about y = 9 y 9 r S 3 )

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