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The Shell Method

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

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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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