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7.3 Integrals and Volume

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Disk Method

We have already seen that integrals canbe used to find areas.

It can also be used to find the volume of

3D shapes On type of solid is that formed by rotating

or revolving a region in a plane about a

line called a solid of revolution We will use the disk methodto find the

volume of such a solid

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Disk Method

Suppose the region formed by y = 3 from

x = 1 to x = 3 is revolved about the x-axis.

The region formed is a disk.

2

V R x

R

x

1

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What if the function is not

constant?

The idea is to partition the interval into

rectangles and revolve the rectangles

Click hereto see illustration.

a b

iR x

x

2

1

n

i

i

V R x x

2

Taking the limit as n givesb

a

V R x dx

http://library.thinkquest.org/3616/Calc/S3/discrep.gifhttp://library.thinkquest.org/3616/Calc/S3/discrep.gif

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Volume of a solid revolution

(disk method) Horizontal axis of rotation:

Vertical axis of rotation:

2

b

a

V R x dx

2

d

c

V R y dy

Horizontal axis of rotation:

Vertical axis of rotation:

Horizontal axis of rotation:

Vertical axis of rotation:

Horizontal axis of rotation:

Vertical axis of rotation:dx

R(x)

R(y)

dy

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Example: Find the volume

Of the region formed by revolving y = .5x

from x = 0 to x = 4 about the x-axis

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Example: Find the volume

Of the region formed by revolving

from x = 0 to about the x-axisx ( ) sing x x

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Example: Find the volume

Of the region formed by revolving

from x = -2 to x = 2 about the x-axis

( ) 2 cosf x x x

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Example: Find the volume

Of the solid formed by revolving the region

between f(x) = 2 x2and g(x) = x about

x = 1

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Example

Find the volume of the solid formed by

rotating the region bounded by

and y = x2about the a) x-axis and b) y-axis

y x

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Example

Find the volume of the solid formed by

revolving from x = 0 to x = 2

about the y-axis

2

2 2

x

y

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Example

Find the volume of the solid formed by

rotating y = x2+ 1 from x=0 to x=1 in the

1stquadrant about the a) y-axis and

b) y = 4

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Example

Find the volume of the solid formed by

rotating the region bounded by y = 4 x2

and y = 0 about

y-axis

y=-3

y=7 x=3

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Known Cross Sections

The base of the disk is a cross section of a

solid formed by revolution

2

2

2

of cross section = R x

( ) R x

R x

( )

b

a

b

a

Area

A x

V dx

A x dx

Therefore,

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Known Cross Sections

In general if the cross section of the solid

is a shape that we know the area of i.e. a

known cross section then

( )b

a

V A x dx If the cross section is

perpendicular to the

x-axis

( )d

c

V A y dy If the cross section is

perpendicular to the

y-axis

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Example

Find the volume of triangular pyramid

w/base formed by f(x) = 1-.5x and

g(x) = -1+.5x and x = 0 with cross sections

equilateral triangles.

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Example

Find the volume of a pyramid 3m high with

a square base that is 3 by 3