VOLUMES
Volume = Area of the base X height
r
x
disk
V   r 2 x
VOLUMES
VOLUMES
V ( Si )  A( xi *)x
For a solid S that isn’t a cylinder we first “cut” S into pieces and approximate
each piece by a cylinder.
V  lim
n 
b
n
 A( x *)x
i 1
V   A( x)dx
a
i
VOLUMES
V   r 2 x
VOLUMES
1 Disk cross-section x
V  lim
n 
b
n
 A( x *)x
i 1
V   A( x)dx
a
i
If the cross-section is a disk, we
find the radius of the disk (in terms
of x ) and use
A   (radius ) 2
Rotating axis
animation
VOLUMES
1 Disk cross-section x
use your imagination
VOLUMES
1 Disk cross-section x
use your imagination
VOLUMES
Intersection point
between L, curve
1 Disk cross-section x
p2
r
step1
step2
Graph and Identify the region
Draw a line perpendicular (L) to the
rotating line at the point x
p1
step3
step4
step5
x
Intersection point
between L, rotating axis
Find the radius r of the disk in
terms of x
Now the cross section Area is
A   r2
Specify the values of x
a xb
step6
The volume is given by
b
V   A( x)dx
a
VOLUMES
VOLUMES
VOLUMES
Volume = Area of the base X height
r
disk
V   r 2 x
x
r1
washer
r2
V   r22 x   r12 x


V   r22  r12 x
x
VOLUMES


V   r22  r12 x
VOLUMES
2 washer cross-section x
V  lim
n 
b
n
 A( x *)x
i 1
V   A( x)dx
a
i
If the cross-section is a washer ,we
find the inner radius and outer radius
A   ( rout ) 2   (rin ) 2
VOLUMES
2 washer cross-section x
Intersection point
between L, boundry
p2
step1
step2
Graph and Identify the
region
Draw a line perpendicular to
the rotating line at the point x
p1
step3
step4
step5
Find the radius r(out) r(in) of
the washer in terms of x
Now the cross section Area is
2
A   (rout
 rin2 )
Specify the values of x
a xb
Intersection point
between L, boundary
step6
The volume is given by
b
V   A( x)dx
a
VOLUMES
T-102
VOLUMES
Example:
Find the volume of the solid obtained by rotating the
region enclosed by the curves y=x and y=x^2 about the
line y=2 . Find the volume of the resulting solid.
VOLUMES
3 Disk cross-section y
V  lim
n 
d
n
 A( y *)y
i 1
V   A( y )dy
c
i
If the cross-section is a disk, we
find the radius of the disk (in terms
of y ) and use
A   (radius ) 2
VOLUMES
3 Disk cross-section y
step1
Graph and Identify the
region
step2
Rewrite all curves as x = in
terms of y
step2
Draw a line perpendicular to
the rotating line at the point y
step3
Find the radius r of the disk
in terms of y
step4
step5
Now the cross section Area is
A   r2
Specify the values of y
c yd
step6
The volume is given by
d
V   A( y )dy
c
VOLUMES
4 washer cross-section y
If the cross-section is a washer ,we
find the inner radius and outer radius
A   ( R) 2   ( r ) 2
Example:
The region enclosed by the curves y=x and y=x^2 is rotated
about the line x=-1 . Find the volume of the resulting solid.
VOLUMES
4 washer cross-section y
step1
Graph and Identify the
region
step2
Rewrite all curves as x = in
terms of y
step2
Draw a line perpendicular to
the rotating line at the point y
step3
Find the radius r(out) and r(in)
of the washer in terms of y
step4
step5
Now the cross section Area is
A   r2
Specify the values of y
c yd
x  1
x y
x
step6
The volume is given by
d
V   A( y )dy
c
y
VOLUMES
4 washer cross-section y
V  lim
n
 A( y *)y
n 
i 1
d
V   A( y )dy
c
i
If the cross-section is a washer ,we
find the inner radius and outer radius
A   ( R) 2   ( r ) 2
T-102
VOLUMES
SUMMARY:
The solids in all previous examples are all called solids of revolution because
they are obtained by revolving a region about a line.
solids of
revolution
NOTE:
solids of
revolution
Rotated by a line
parallel to x-axis ( y=c)
Rotated by a line
parallel to y-axis ( x=c)
b
V   A( x)dx
a
d
V   A( y )dy
c
The cross section is perpendicular to the rotating line
Cross-section is
DISK
Cross—section is
WASHER
A   (R) 2
A   ( R) 2   ( r ) 2
VOLUMES
Sec 6.2: VOLUMES
not solids of revolution
The solids in all previous examples are all called solids of revolution because
they are obtained by revolving a region about a line.
We now consider the volumes of solids that are not solids of revolution.
VOLUMES
T-102
VOLUMES
T-092