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July 6th, 2012, 08:14 PM   #1
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Difference between Disk/Washer and Cylindrical Shells

Just a question I have about these two methods of finding volumes of revolution.
So, the formula for Disk/Washer is: Pi*Integral(f(x)^2dx))
The formula for Cyllindrical Shells is: 2*Pi*Integral(x*f(x)dx))
Do these formulae give equivalent results for every function in every interval?
So, suppose we want to find a volume of revolution of the function y = x from 3 to 2 about the x-axis.
We have: V = Pi*Integral(x^2dx)from 2 to 3 = Pi/3 * 27 - Pi/3 * 8 = 19.8968 u^3
While, if we use the Cyllindrical Shells method, we will get the doubled answer..
I am pretty confused about this, but maybe I am doing something wrong, so does anyone mind explaining?
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July 6th, 2012, 08:37 PM   #2
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Re: Difference between Disk/Washer and Cylindrical Shells

You will always get the same result regardless of the method you choose to find the volume of a solid of revolution.

Using you example, we find:

Disk method:



Shell method:

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July 6th, 2012, 09:03 PM   #3
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Re: Difference between Disk/Washer and Cylindrical Shells

Right, it's because the functions of x is used in cylindrical shells method when it is revolving about the y-axis, not the x-axis. When I want to use the cylindrical shells method to find the volume of revolution about the x-axis, I need to rearrange everything in terms of y and then integrate. Correct?
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July 6th, 2012, 09:25 PM   #4
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Re: Difference between Disk/Washer and Cylindrical Shells

Yes. Often one method is easier to compute than the other.
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July 6th, 2012, 09:36 PM   #5
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Re: Difference between Disk/Washer and Cylindrical Shells

All right, thanks, that clears all the confusion.
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