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/