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### Calculus - Shell or Washer Method?

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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

1) y = 0 (x-axis)
2) x = 0 (y-axis)
3) x = 2
4) y = -3

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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

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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Associated Topics:
High School Calculus

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