When to use which method of finding volumes

When it's clear that, when using the disk/washer method, you will be able to define the function variable in terms of the differentiation variable, you can use that method.

Illustration: the way to calculate volume with the disk method is pi∫(A to B)[F(x)^2-f(x)^2]dy , requiring a translation of [F(x)^2 - f(x)^2] into terms of y; finding F inverse and f inverse.

Otherwise, you should use the cylindrical shells, which have integration in terms of the same variable as the function inside.

Click on one of the "Math Notes" pictures, and then click "View All" after it sends you to Picasa, and you will see that one of the pictures shows the table that Prof. Newberry set up, showing when to use the disk/washer method and when to use the cylindrical shell method.

The idea of the disk method is ∫pi(radius squared)(thickness), or pi ∫ f(x)^2 dx .

When there is a hole in the middle, the washer method is used: 

∫ [pi (big radius squared) - pi (little radius squared)] (thickness) , 

or pi ∫ [F(x)^2 - f(x)^2] dx .

The cylindrical shell method fully takes care of whether and where there is a hole using the limits of the integration, which show where the outermost and innermost radii end. 

The integral is written: 

∫ (circumference)(height)(thickness) , 

or ∫ (2 pi radius)(F(x) - f(x)) dx , 

or 2 pi ∫ ( x - [axis] )( F(x) - f(x) ) dx