Comparing washer and shell method

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We compare and contrast the washer and shell method.

Bringing it all together

We have seen two different techniques that can be used to find the volume of a solid of revolution. We summarize the washer and shell method side by side.

	Washer Method	Shell Method


Orientation of slices:	perpendicular to the axis	parallel to the axis
For vertical slices:	$V=\int _{x=a}^{x=b} \pi (R^2-r^2) \d x$	$V=\int _{x=a}^{x=b} 2\pi \rho h \d x$
For horizontal slices:	$V=\int _{y=c}^{y=d} \pi (R^2-r^2) \d y$	$V=\int _{y=c}^{y=d} 2\pi \rho h \d y$
Geometric quantities:	$R$ - outer radius	$\rho$ - shell radius
	$r$ - inner radius	$h$ - length of slice

We find the geometric quantities by noting the following.

The outer radius $R$ is the distance from the axis of revolution to the outer curve.
The inner radius $r$ is the distance from the axis of revolution to the inner curve.
The shell radius $\rho$ is the distance from the axis of rotation to the representative slice.
The length $h$ is the height of a vertical slice or the width of a horizontal slice.

Given a region of revolution and an axis of revolution there are three important pieces of information that ultimately must be considered to set up an integral or sum of integrals that gives the volume of the corresponding solid of revolution.

The variable of integration ( $x$ or $y$ )
The method (washer or shell)
The type of slice (vertical or horizontal)

An important observation is that given any one of these three pieces of information, the others immediately follow. Here are a few examples.

The region bounded by $x=\frac {2}{y}$ , $y=-2$ , $x=4$ and $x=9$ is revolved about the $y$ -axis. If an integral or sum of integrals with respect to $y$ is used to compute the volume of the solid, should we use the washer or the shell method?

To answer this, note that since we integrate with respect to $y$ , we use horizontalvertical slices. These slices are parallelperpendicular to the axis of rotation. Thus, we should use the washershell method.

Here is an example in which we are given the method.

The region bounded by $y=1-x$ , $x=0$ , and $y=0$ is revolved about the line $x=2$ . If the washer method is used to calculate the volume or the resulting solid, should we integrate with respect to $x$ or $y$ ?

Since we use the washer method, the slices must be parallelperpendicular to the axis of rotation. These slices are verticalhorizontal , so we should integrate with respect to $x$ $y$ .

Sometimes, we only want to find the volume of a solid of revolution and we are not given the method nor the variable of integration. In this case, choosing the type of slice that is more convenient to use in the region of revolution is a good idea.

The region in the $xy$ -plane bounded between $x=1$ , $y=2x-3$ , and $y=5+\frac {8}{x-4}$ is shown below.

A solid of revolution is formed by revolving this region about $y=3$ . What is the variable of integration and which method should be used to find the volume?

If we use horizontal slices, we will need two integrals but if we use vertical slices, we will only need one. We thus choose to use vertical slices. This means that our variable of integration is $x$ $y$ . Now, notice that vertical slices are perpendicularparallel to the axis of rotation $y=3$ . Thus, we should use the washershell method.

“Education is not the learning of facts, but the training of the mind to think” - Albert Einstein