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The Shell Method

Learn how to find volumes by cylindrical shells method, revolutionize solid shapes around the y-axis with examples and step-by-step calculations. Understand the concept and formula using practical applications.

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The Shell Method

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  1. The Shell Method Volumes by Cylindrical Shells By Christine Li, Per. 4

  2. Conceptually the same as the washer method • Do not sum up the volumes of the washers • Instead of summing the volumes of washers you sum up the volumes of shells

  3. Example 1 • Question: • Find the volume of the solid obtained by rotating about the y-axis the region bounded by y = x2 and between x = 1 and x = 2

  4. Example 1 First consider the small vertical strip below with width dx And then revolve it around the y-axis to get As you can see, the shape that the strip sweeps out resembles a shell

  5. Example 1 • The next step is to calculate the volume of this single shell. Since the thickness is dx, all you need is to calculate the surface area of the shell • volume = thickness x surface area

  6. <-----(2pi)r----> Example 1: Surface Area of a Shell • First, cut the shell and unroll it, and get the following picture h • Since (2pi)r is the length, h is the width, and dx is the depth, the volume of this shell is (2pi)rh dx.

  7. <-----(2pi)r----> Example 1:But what are r and h? • We see that the distance from the y-axis to the strip is x • Therefore, r = x • Since the height of the strip is the distance from the x-axis to the curve (y = x2) we see that h = x2 • The volume of a single shell is (2pi)x3 dx h

  8. Example 1:Approximating the volume • Now approximate the volume of the solid of revolution by summing the volumes of all the shells from x = 1 to x = 2 • We then take the limit as the number of shells tends to infinity • to find the volume of the solid of revolution

  9. Example 1: Final Step • The limit converges to the following integral 2 2pi x3 dx = (15pi)/2 1 • In general, we have the following formula for the volume of a solid of revolution using the shell method : Where r is the radius of the shell, h is the height of the shell and a and b are the bounds on the domain of x. b (2pi)rh dx a

  10. Example 2 • Question: • Find the volume of the solid of revolution when the region below y = x + 1 and between x = 0 and x = 2 is revolved about the line x = 3 revolved around x = 3 and get

  11. Example 2:But what are r and h? • Here we take a vertical strip with thickness dx to get a shell • The radius is the distance from the curve ( y = x + 1) to the line x = 3 • The radius is r = 3 - x (remember the radius has to be in terms of the variable x). • The height is the distance from the x-axis to the curve ( y = x + 1) • Since y = x + 1, the height of the strip is h = x + 1

  12. = (44pi)/3 Example 2: Final Step • Use the shell method formula • and remember that the area is bounded between x = 0 and x = 2 • Therefore, set 0 to be the lower limit, and 2 as the upper limit. 2 (2pi)(3 - x)(x + 1) dx 0

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