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

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

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

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 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 Example 1

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.

Example 1: But what are r and h?  Since the height of the strip is the distance from the x-axis to the curve (y = x 2 ) we see that h = x 2  The volume of a single shell is (2pi)x 3 dx h  We see that the distance from the y-axis to the strip is x Therefore, r = x

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

 The limit converges to the following integral 2 1 2pi x 3 dx = (15pi)/2  In general, we have the following formula for the volume of a solid of revolution using the shell method : b a (2pi)rh dx 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. Example 1: Final Step

 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 Example 2 revolved around x = 3 and get

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

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. (2pi)(3 - x)(x + 1) dx 2 0 =(44pi)/3