Editing Shell integration (section)

==Example==

Consider the volume, depicted below, whose cross section on the interval [1, 2] is defined by:

:<math>y = 8(x-1)^2(x-2)^2</math>

{{Multiple image
 | align = none
 | total_width = 600
 | image1 = Shell_2d_example.png
 | width1 = 481 | height1 = 307
 | caption1 = Cross-section
 | image2 = Shell_3D_example.png
 | width2 = 632 | height2 = 463
 | caption2 = 3D volume
 | caption_align = center
}}

With the shell method we simply use the following formula:

:<math>V = 16 \pi \int_1^2 x ((x-1)^2(x-2)^2) \,dx </math>

By expanding the polynomial, the integration is easily done giving {{sfrac|8|10}}<math>\pi</math> cubic units.

=== Comparison With Disc Integration ===

Much more work is needed to find the volume if we use [[disc integration]].  First, we would need to solve <math>y = 8(x-1)^2(x-2)^2</math> for {{mvar|x}}.  Next, because the volume is hollow in the middle, we would need two functions: one that defined an outer solid and one that defined the inner hollow. After integrating each of these two functions, we would subtract them to yield the desired volume.