1. 2. Warm-Up: 1. Warm-Up: 2. Warm-Up: HW ? 7-2: Volume of Solids of Revolution Find volume using the disk and washer methods Find volume of solids.

1.

2.

Warm-Up:

1.

Warm-Up:

2.

Warm-Up:

HW ?

7-2: Volume of Solids of Revolution

•Find volume using the disk and washer methods•Find volume of solids with known cross sections

©2002Roy L. Gover (www.mrgover.com)

Objectives:

Examples

Created by Lawrence Hunsh

Example2y x from [0,2]

Rotate about x axis

Creates

Partition with disks

Important Idea

xy

Each cylinder is a disk

Solid of Revolution

Important Idea

Volume of ea. disk= ri

2x

ri

x

R

2 2i iR xr

The total volume is given by the Riemann Sum:

2

01

limn

ix

i

v r x

2( )

b

ar x dx

The definite integral is the accumulator of the disk volumes

The Disk MethodTo find the volume of a solid of revolution, use one of the following:

Hori. Axis of Rev.

Vert. Axis of Rev.

2

1

2( )

x

xv r x dx

2

1

2( )

y

yv r y dy

ExampleFind the volume of the solid of revolution formed by revolving the graph:

around the x axis. R is a constant

2 2y R x

Assignment1. 463/1-4 all,

Slides 1-15

Slides 16-23

2. 464/9-12 all 37,39,43

3. 463/7,8,13,17,19-23 odd, 51,52

Warm-UpFind the volume of the

solid of revolution formed by revolving the graph around

thefrom x=0 to x=1x axis.

Show your integralsetup and evaluate with

your calculator. Your answer should be accurate to 3 decimal places.

xy e

Solution

0 1 Setup

Top half of Disks

21

010.036 cu. un.xv e dx

Try ThisFind the

volume of the solid of revolution formed by revolving the grapharound the x axis.

0

cu. unitscosy x

2

ExampleFind the

volume of the solid of revolution formed by revolving the grapharound the y axis.

0 1

2y x

Try ThisFind the

volume of the solid of revolution formed by revolving the

0 2region bounded by y=2x2, x=0 & x=2 about the y axis.

16 cu. un.

ExampleFind the

volume of the solid formed by revolving

the region bounded by the graphs

y=2x2, y=0 & x=2 about the line x=2.

8

0 2

Try ThisFind the volume of the solid formed by revolving

the region bounded by the graphs

y=x2, y=0 & x=3 about the line x=3.

0 3

27 cu

2.. un

The Washer MethodIf the disk has a hole, it is a washer

Rr

2 2( )R xr Vol. Of washer

x

Important IdeaThe volume of a solid of revolution with a hole is the volume of the solid without the hole less the volume of the hole.

ExampleFind the

volume of the solid formed by revolving

the region bounded by the graphs

y=x and y=x2 about the x axis.

Rr

ExampleFind the

volume of the solid formed by revolving

the region bounded by the graphs

y=x and y=x2 about the y axis.

R

r

Assignment1. 463/1-4all

Slides 16-23

2. 463/7,8,13,17, 37,39,43

Warm-UpFind the volume of the solid generated when the region between the graphs f(x)=1/2over the interval

[0,2]is revolved about the x axis.

+x2 andg(x)=x

69 cu. un.

10

ExampleFind the

volume of the solid formed by revolving

the region bounded by the graphs

y=2x2, y=0 & x=2 about the line y=8.

8

0 2

Try ThisFind the volume of the solid formed by revolvingthe region bounded by the graphs , y=0 & x=1, x=4, about the line y=4.

1 40

4

1y

x

Solution

1 40

4

24 2

1

14 4v dx

x

38ln 4

4

32.485

Solids with Known Cross Sections

x

y

( )b

av A x dx

ExampleFind the volume of the solid whose base is bounded by the circle x2+y2=4 with equilateral triangle cross

sectionsperpendicular to the x axis.

x y

Try ThisFind the volume of the solid whose base is bounded by the circle x2+y2=4 with semicircular cross

sectionsperpendicular to the x axis.

x y

16 cu

3.. un

Lesson CloseWrite a paragraph describing how you find the volume of a solid of revolution.Dr. Lou Talman, Metro State University, Denver, CO. "Solids of Revolution." [Online image] 29 December 2004.<httpp://curvebank/calstatela.edu/volrev/volrev.htm>.Unknown Author,"Solids of Revolution." [Online image] December 2004.<http://chuwm2.tripod.com/revolution/volume.html>.Lawrence S. Husch, University of Tenn., Knoxville. "Visual Calculus-Solids of Revolution." [Online image] 29 December 2004.<http://archives.math.utk.edu/visual.calculus/5/volumes.5/index.html>.

Credits-animated pictures on slides 1,2 and 3

Assignment1. 463/1-6 all, 37,39,43

Slides 1-15

Slides 16-23

2. 464/9-12 all3. 463/7,8,13,17,19-23 odd, 51,52

(Disks about y Axis)