To start, read: When to use Disk Method versus Shell Method, Part 1 to get a general visual. This is a little more detailed:

Volume should be thought of as infinitely stacked area. In the disk method, you are infinitely stacking circles (think pancakes). In the shell method, you are infinitely stacking lateral surface areas of cylinders (think “Russian Dolls” that stack inside of each other)

The Disk Method: Since you are stacking pancakes, the general formula that you will be integrating is π*r^{2}. If the radius of each disk is changing throughout the shape, the radius, r, will be a function, dependent on either y or x, depending on how you are rotating.

The Shell Method: Since you are stacking lateral areas of cylinders, the general formula that you will be integrating is 2*π*r*h (lateral area of cylinder formula). The radius, r, will be a simple function involving an x or a y. The height, h, will depend on the functions that are being rotated.

Now that you have an understanding of the concept, view this example… I solve a volume problem using the disk method. Then, I solve the exact same problem using the shell method: Integration Example: Disk Method vs. Shell Method

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