Calculus - Shell or Washer Method?
Date: 08/23/97 at 19:12:33 From: Tracy Subject: Calculus Dear Dr. Math, I have a question regarding how to set up integrals for the volume of revolution. The major things that I am not certain about are: 1) How do I know when to use the "shell" method or the "washer" method (is the "disc" method one of those? and 2) I am not sure how to set up the problem when it is revolved around the y-axis instead of the x-axis. Do I put in y values for the integral or do I still use x values? Well, here is the problem: Let R be the region bounded by the curves y = x^3, y = 0, and x = 2. Set up definite integrals for the volume of revolution obtained by revolving R about the line: 1) y = 0 (x-axis) 2) x = 0 (y-axis) 3) x = 2 4) y = -3 Thanks for your help!
Date: 08/29/97 at 14:10:58 From: Doctor Rob Subject: Re: Calculus Either method will work if properly applied. There are no problems I know of where one will and the other won't. Probably you would choose the one which made expressing y as a function of x or x as a function of y the simplest. If you have y as a function of x, then rotating about the x-axis would give you a disk/washer method, and about the y-axis would give you a shell method. If you have x as a function of y, the reverse would be true. The formula using the disk or washer method rotating about the x-axis is b V = Int Pi*(f(x)^2-g(x)^2) dx a where y = f(x) is the outer boundary and y = g(x) is the inner boundary in the y-direction, expressed as functions of x, and x ranges from a to b. The formula using the shell method rotating about the x-axis is b V = Int 2*Pi*(F(y)-G(y))*y dy a where x = F(y) is the outer boundary in the x-direction and x = G(y) is the inner boundary in the x-direction, expressed as functions of x, and y ranges from a to b. To rotate about the y-axis, swap the roles of x and y throughout the above discussion. To rotate about a different axis, change coordinates so that the axis of rotation is one of the coordinate axes, and then do the computation. In your example, you let R be the region bounded by the curves y = x^3, y = 0, and x = 2. You want to find the volume revolving R about the lines: 1) y = 0 (x-axis) 2) x = 0 (y-axis) 3) x = 2 4) y = -3 1) Since you have y expressed as a function of x already, you should probably use the disk/washer method. f(x) = x^3, g(x) = 0, a = 0, b = 2. 2) Since you have y expressed as a function of x already, you should probably use the shell method. F(x) = x^3, G(x) = 0, a = 0, b = 2. 3) Make the substitution X = x-2, or x = X+2. Then the axis of rotation is X = 0, the y-axis in the Xy-coordinate system. Use the shell method with F(X) = (X+2)^3, G(X) = 0, a = -2, b = 0. 4) Make the substitution Y = y+3, or y = Y-3. Then the axis of rotation is Y = 0, the x-axis in the xY-coordinate system. Use the disk/washer method with f(x) = x^3 + 3, g(x) = 3, a = 0, b = 2. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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