Volume

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. Axel - Funnel media type="custom" key="3817319"media type="custom" key="3817327" For your entertainment:)

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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 revolu tion.

 media type="youtube" key="F2psxMnGdUw" height="344" width="425" Video

The equation for the disk method is given by

[[image:disk.jpg width="226" height="74"]]
 **//EXAMPLE//** Find the volume of the solid obtained by rotating about the x -axis the region under the curve y = from 0 to 1. Solution. From our formula, taking a = 0 and b = 1, and f ( x ) = :

Washer Method
The disk method can be used to find the volume of solids with holes by replacing the disk with a **washer**





(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).**

<span style="font-size: 11pt; line-height: 115%; font-family: 'Calibri','sans-serif';">The volume of the resulting solid is given by

[[image:washer.jpg width="396" height="82"]]
<span style="font-family: Verdana,Geneva,sans-serif;">**EXAMPLE** The region bounded by the curves <span style="font-family: Verdana,Geneva,sans-serif;"> y = sin( x ), y = cos( x ) and y = 0 is rotated around the x -axis. Find the volume of the resulting solid. The curves y = sin( x ) and y = cos( x ) meet at x = p i/ 4, with common value 1 /. 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:
 * Solution**

This is equal because of a trig identity

<span style="font-family: Verdana,Geneva,sans-serif;">And the volume of S is: <span style="display: block; font-family: Verdana,Geneva,sans-serif; text-align: center;"> 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:

<span style="display: block; font-size: 210%; color: rgb(0, 0, 255); text-align: center;">Shell Method
<span style="font-family: Verdana,Geneva,sans-serif;">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. <span style="font-size: 12pt; font-family: 'Times New Roman','serif';"> Matroyska dolls showing shell concept

<span style="font-family: Verdana,Geneva,sans-serif;"><span style="font-size: 12pt; font-family: 'Times New Roman','serif';">Mathematically, this method is represented by: <span style="font-size: 12pt; line-height: 115%; font-family: 'Times New Roman','serif';">

if the rotation is around the x-axis (horizontal axis of revolution), or <span style="font-size: 12pt; line-height: 115%; font-family: 'Times New Roman','serif';"> <span style="display: block; font-size: 12pt; font-family: 'Times New Roman',Times,serif; text-align: center;"><span style="font-family: Verdana,Geneva,sans-serif;">if the rotation is around the y-axis (vertical axis of revolution).

Find the volume of the solid generated by revolving the region bounded by //y// = //x//2 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//) = //x//2
 * 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** media type="youtube" key="NIdqkwocNuE" height="344" width="425" Video <span style="font-family: Verdana,Geneva,sans-serif;"> **EXAMPLE**



<span style="font-family: Verdana,Geneva,sans-serif;">The volume ( //V//) of the solid is



<span style="display: block; font-size: 210%; color: rgb(0, 0, 255); text-align: center;">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

<span style="display: block; font-size: 210%; color: rgb(0, 0, 255); text-align: center;">Works Cite<span style="font-family: Arial,Helvetica,sans-serif;">d
<span style="font-size: 140%; color: rgb(128, 0, 128); background-color: rgb(255, 255, 0);"> 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.