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## AP®︎ Calculus AB (2017 edition)

### Course: AP®︎ Calculus AB (2017 edition)>Unit 11

Lesson 8: Volume: shell method (optional)

# Shell method for rotating around vertical line

Introducing the shell method for rotation around a vertical line. Created by Sal Khan.

## Want to join the conversation?

• I need some help with the conceptual part. Conceptually, these shells stack INTO each other, kind of like a Russian Porcelian doll set right? Whereas the disc method you are simply laying discus on top of each other, kind of like laying pizzas on top of each other.
• That is a valid visualization of the two operations.
• I'm having a confusion that outer surface area of a cylinder is 2pi*r*h........from where that dx came from...........plz can you explain in terms of cylinder
• The surface area of a cylinder has zero thickness, so it can't be used to create something that has any volume. For a volume calculation, we need something with at least a little thickness, and in this case the small increment of thickness is in the horizontal direction, so we call it dx. It's as if we made a cylindrical shell by rolling up a piece of paper. The volume of that shell would be the surface area of the paper (2πrh) times the thickness of the paper (dx).
• So when do you use the shells method as opposed to the washers method?
• Whichever makes evaluating the integral easier for you - you are free to choose.
• I get that multiplying the surface area of a cylinder by a very small thickness would approximate the volume of that very thin cylinder. But it seems that that remains always an approximation since the inner surface of a cylinder will always have a smaller circumference than the outer surface. So the result would have to be an over or under approximation, especially if the calculation is repeated. The discrepancy get smaller but the repetitions increase. Am I missing something?
• is the radius always going to be x? no matter the function or interval?
• How do I know when to use the shell method over the disk/washer method??
• if your function is hard to define explicitly in terms of y, so like this function (x-1)(x-3)^2 is hard to write in terms of y
(1 vote)
• I'm still confused on how to find the radius whenever it isn't on one of the axes, can someone help me?
• The shell looks like the washer method-How are they different?
• It's basically the same as the washer method, but you want to use this method when it's difficult to express an equation as a function of y or x, as Sal said at .
• I am not sure, but is the volume of the shell equal to the volume of the parallelepiped?? I think the volume of the shell must be equal to the volume of the bigger cylinder mines the volume of the smaller cylinder.