Volume
By: Kevin Haigler and Sarah Cashdollar
Volume, along with Area, is one of the most common applications of the integral. We will study how to find the volume of three-dimensional solids whose cross sections are similar. These solids of revolution, such as axles, funnels, bottles, and pistons, are commonly used in engineering and manufacturing.

axle.jpgfunnel.jpg
Axel - Funnel


For your entertainment:)




Commonly Used Terms

Solid of Revolution - Region in a plane that is revolved around a line
Axis of Revolution- line about which solid is revolved
Cross Section- "slice" of solid of revolution perpendicular to the axis of revolution
Disk- revolved solid that has one side adjacent to the axis of revolution
Washer- revolved whose side closest to the axis of revolution is not touching the axis

Disk Method

The simplest type of solid is the disk, or right circular cylinder, which is formed by revolving a rectangular cross section about an axis adjacent to one side of the rectangle. This cross section is perpendicular to the axis of revolution.

Video

The equation for the disk method is given by

disk.jpg

EXAMPLE
Find the volume of the solid obtained by rotating about the
x-axis the region under the curve y = cubedrootx.giffrom 0 to 1.
Solution. From our formula, taking a = 0 and b = 1, and f(x) =cubedrootx.gif :

disk_example.gif


Washer Method

The disk method can be used to find the volume of solids with holes by replacing the disk with a washer
Aluminium discs by Katie-Rose.
Aluminium discs by Katie-Rose.

down_arrow.png


external image Yogurt_Burst_Cheerios.PNG


(Disks to Washers)

Like with disks, the cross section is perpendicular to the axis of revolution.
Consider a region bounded by an outer radius R(x) and an inner radius r(x).

washer.gif

The volume of the resulting solid is given by

washer.jpg


EXAMPLE
The region bounded by the curves
y = sin(x), y = cos(x) and y = 0 is rotated around the x-axis. Find the volume of the resulting solid.
Solution
The curves y = sin(x) and y = cos(x) meet at x = pi/4, with common value 1/root2.gif. Let a = 0, b = p/4. Let R be the region, and S the solid of revolution.
A cross section in the plane P x of S is an annulus (or a washer) with inner radius sin(x) and outer radius cos(x).
Hence the cross sectional area is:


WASHER1.gif
This is equal because of a trig identity



And the volume of S is:
WASHER2.gif
If a solid of revolution S is created by: rotating a plane region bounded by the curves y = f(x), y = g(x), x = a and x = b, where f(x) ³ g(x) on [a, b], about the x-axis.Then by the above washer method, the volume is:
WASHER3.gif




Shell Method

An alternative method for finding volumes of solids is the shell method, which uses cylindrical shells.
As opposed to the washer method, with the shell method the rectangular cross section is horizontal to the axis of revolution.


dolls.jpg
Matroyska dolls showing shell concept

Mathematically, this method is represented by:

if the rotation is around the x-axis (horizontal axis of revolution), or

shell_x.jpg

if the rotation is around the y-axis (vertical axis of revolution).

Where the function p(x) is the distance from the axis and h(x) is the height of the shell, generally the region being rotated. The values for a and b are the limits of integration, the starting and stopping points of the rotated shape
Video EXAMPLE
Find the volume of the solid generated by revolving the region bounded by y = x2 and the x-axis [1,3] about the y-axis.
In using the cylindrical shell method, the integral should be expressed in terms of x because the axis of revolution is vertical. The radius of the shell is x, and the height of the shell is f(x) = x2


SHELL.jpg




The volume ( V) of the solid is

Shells2.gif

Need Extra Practice...


...with Disk Method?
http://archives.math.utk.edu/visual.calculus/5/volumes.5/
...with Washer Method?
http://cow.temple.edu/~cow/cgi-bin/manager calculus book 2, 2 Applications of Integration, 2 Volume, 1 Solids of Revolution- Washers
...with Shell Method?
http://cow.temple.edu/~cow/cgi-bin/manager calculus book 2, 2 Applications of Integration, 2 Volume, 2 Solids of Revolution-Shells
...all?
http://www.intmath.com/Application-integration/4_Volume-solid-revolution.php

Works Cited


Flickr
Google Images
Wikipedia
CliffsNotes.com. Volumes of Solids of Revolution. 18 May 2009
<http://www.cliffsnotes.com/WileyCDA/CliffsReviewTopic/topicArticleId-39909,articleId-39907.html>.
Larson, Ron, Hostetler, Robert P., Edwards, Bruce. Calculus of a Single Variable. 2006. Boston; Houghton Mifflin.