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33
3
Find the volume of the pyramid:
Consider a horizontal slice through the pyramid.
sdh
The volume of the slice is s2dh.
If we put zero at the top of the pyramid and make down the positive direction, then s=h.
0
3
h
2slice
V h dh=
32
0V h dh= ò
3
3
0
13h= 9=
This correlates with the formula:13
V Bh=1
9 33
= × × 9=
Method of Slicing:
1
Find a formula for A(x).
Sketch the solid and a typical cross section.
2
3 Find the limits of integration.
4 Integrate V(x) to find volume.
®
Volume formula :: Let S be a solid bounded by two parallel planes perpendicular to the x-axis at x=a and x=b. If, for each x in [a,b], the cross-sectional area of S perpendicular to the x-axis is A(x), then the volume of the solid is
( )b
a
V A x dx=ò
provided A(x) is integrable.
Volume formula :: Let S be a solid bounded by two parallel planes perpendicular to the y-axis at y=c and y=d. If, for each y in [c,d], the cross-sectional area of S perpendicular to the y-axis is A(y), then the volume of the solid is
( )d
c
V A y dy= ò
provided A(y) is integrable.
xy
A 45o wedge is cut from a cylinder of radius 3 as shown.
Find the volume of the wedge.
You could slice this wedge shape several ways, but the simplest cross section is a rectangle.
If we let h equal the height of the slice then the volume of the slice is: ( ) 2= ××V x y h dx
Since the wedge is cut at a 45o angle:x
h45o h x=
Since 2 2 9x y+ = 29y x= -®
xy
( ) 2V x y h dx= × ×h x=
29y x= -
( ) 22 9V x x x dx= - × ×
3 2
02 9V x x dx= -ò
29u x= -2 du x dx= -
( )0 9u =( )3 0u =
10
29
V u du=-ò9
32
0
23u=
227
3= × 18=
Even though we started with a cylinder, does not enter the calculation!
®
Cavalieri’s Theorem:
Two solids with equal altitudes and identical parallel cross sections have the same volume.
Identical Cross Sections
y x= Suppose we start with this curve.
If we want to build a nose cone in this shape.
So we put a piece of wood in a lathe and turn it to a shape to match the curve.
®
y x=How could we find the volume of the cone?
One way would be to cut it into a series of thin slices (flat cylinders) and add their volumes.
The volume of each flat cylinder (disk) is:
2 the thicknessrp ×
In this case:
r= the y value of the function
thickness = a small change
in x = dx
( )2
x dx
®
y x=
The volume of each flat cylinder (disk) is:
2 the thicknessrp ×
If we add the volumes, we get:
( )24
0x dxpò
4
0 x dxp= ò
4
2
02x
p= 8p=
( )2
x dx
®
This application of the method of slicing is called the disk method. The shape of the slice is a disk, so we use the formula for the area of a circle to find the volume of the disk.
If the shape is rotated about the x-axis, then the formula is:
A shape rotated about the y-axis would be:
®
2 2( )
b b
a a
V r dx f x dxp pé ù é ù= =ë û ë ûò ò
2( )
d
c
V gy dypé ù= ê úë ûò
Find the volume of the solid generated by revolving the regionsabout the x-axis.2y x x and y 0= - =bounded by
12
0
r dxpò ( )1
22
0
dx xx= -pò
Find the volume of the solid generated by revolving the regionsabout the x-axis.2y 3x x and y 0= - - =bounded by
02
3
dxr-
pò ( )20
2
3
dx3x x-
- -= pò
Find the volume of the solid generated by revolving the regionsabout the y-axis.1
2y x, x 0, y 2= = =bounded by
22
0
r dypò ( )2
2
0
d2y y= pò
Find the volume of the solid generated by revolving the regionsabout the x-axis.2
y 2sin2x, 0 x p= £ £bounded by
/ 22
0
xr dp
pò ( )/ 2
2
0
dx2sin2xp
= pò
Find the volume of the solid generated by revolving the regionsabout the line y = -1.
2y 3 x , y 1= - = -bounded by
22
2
dxr-
pò ( ) ( )( )2 2
2
2
3 x 1 dx-
-= - -pò
The region between the curve , and the
y-axis is revolved about the y-axis. Find the volume.
1x
y= 1 4y£ £
y x
1 1
2
3
4
1.707
2=
1.577
3=
1
2
We use a horizontal disk.
dy
The thickness is dy.
The radius is the x value of the function .1
y=
24
1
1 V dy
yp
æ ö÷ç ÷ç= ÷ç ÷ç ÷çè øò
volume of disk
4
1
1 dy
yp= ò
4
1lnyp= ( )ln4 ln1p= -
02ln2p= 2 ln2p=
®
The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of this equation revolved about the y-axis:
2.000574 .439 185x y y= - +x
y
500 ft
( )500 2
2
0.000574 .439 185 y y dyp - +ò
The volume can be calculated using the disk method with a horizontal disk.
324,700,000 ft»
The region bounded by and is revolved about the y-axis.Find the volume.
2y x= 2y x=
The “disk” now has a hole in it, making it a “washer”.
If we use a horizontal slice:
The volume of the washer is: ( )2 2 thicknessR rp p- ×
( )2 2R r dyp -
outerradius
innerradius
2y x=
2y
x=
2y x=
y x=
2y x=
2y x=
( )2
24
0 2y
V y dypæ öæö÷ç ÷÷çç ÷÷= - çç ÷÷çç ÷ç ÷è øç ÷çè ø
ò
42
0
14
V y y dypæ ö÷ç ÷= -ç ÷ç ÷çè øò
42
0
1
4V y y dyp= -ò
4
2 3
0
1 12 12y yp
é ùê ú= -ê úë û
168
3p
é ùê ú= -ê úë û
83p
=®
This application of the method of slicing is called the washer method. The shape of the slice is a circle with a hole in it, so we subtract the area of the inner circle from the area of the outer circle.
The washer method formula is:
®
( )2 2
2 2( ) ( )
b
ab
a
V R r dx
f x g x dx
p
p
= -
æ öé ù é ù= -ç ÷ë û ë ûè ø
ò
ò
2y x=If the same region is rotated about the line x=2:
2y x=
The outer radius is:
22y
R = -
R
The inner radius is:
2r y= -
r
2y x=
2y
x=
2y x=
y x=
42 2
0V R r dyp= -ò
( )2
24
02 2
2y
y dypæ ö÷ç ÷= - - -ç ÷ç ÷çè øò
( )24
04 2 4 4
4y
y y y dypæ ö÷ç ÷= - + - - +ç ÷ç ÷çè ø
ò
24
04 2 4 4
4y
y y y dyp= - + - + -ò14
2 2
0
13 4
4y y y dyp= - + +ò
43
2 3 2
0
3 1 82 12 3y y yp
é ùê ú= ×- + +ê úê úë û
16 6424
3 3p
é ùê ú= ×- + +ê úë û
83p
=