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 July 6th, 2012, 09:14 PM #1 Newbie   Joined: Feb 2012 Posts: 9 Thanks: 0 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?
 July 6th, 2012, 09:37 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 464 Math Focus: Calculus/ODEs 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: $V=\pi\int_2\,^3x^2\,dx=\frac{\pi}{3}$x^3$_2^3=\frac{19\pi}{3}$ Shell method: $V=2\pi$$\int_0\,^2 y\,dy+\int_2\,^3 y(3-y)\,dy$$=2\pi$$\frac{1}{2}$y^2$_0^2+$\frac{3}{2}y^2-\frac{1}{3}y^3$_2^3$$=2\pi$$2+\frac{7}{6}$$=\frac{19\pi}{3}$
 July 6th, 2012, 10:03 PM #3 Newbie   Joined: Feb 2012 Posts: 9 Thanks: 0 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?
 July 6th, 2012, 10:25 PM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 464 Math Focus: Calculus/ODEs Re: Difference between Disk/Washer and Cylindrical Shells Yes. Often one method is easier to compute than the other.
 July 6th, 2012, 10:36 PM #5 Newbie   Joined: Feb 2012 Posts: 9 Thanks: 0 Re: Difference between Disk/Washer and Cylindrical Shells All right, thanks, that clears all the confusion.

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