433
Answers
601–700
Answers and Explanations
693.
4 096
9
,
π
To find the volume of a solid obtained by rotating about the x-axis a region to the left of
the curve x = f
 
(y) and to the right of the y-axis from y = c to y = d with cylindrical shells,
use the integral
V
yf y dy
c
d
= ∫ 2π ( )
More generically, you can use the formula
d
V
y
c
d
= ∫
2π (
) (
)
shell radius shell height
To find the shell radius, let y be in the interval [c, d]; the shell radius is the distance
from y to the line of rotation. To find the shell height, let f
 
(y) be the curve that bounds
the region on the right and let g(y) be the curve that bounds the region on the left; the
shell height is given by f
 
(y) – g(y). Note that the limits of integration often correspond
to points of intersection.
Begin by isolating the y in the first equation to get y
1/4
= x. Find the lower limit of integration by solving y
1/4
= 0 to get y = 0. Then find the volume using cylindrical shells:
2
2
2 4
9
2 4
9
16
0
4 096
1 4
0
16
5 4
0
16
9 4
0
16
9 4
π
π
π
π
y y
dy
y dy
y
( ) =
=
=
−
(
)
=
∫
∫
( )
,
π π
9
694.
625
2
π
To find the volume of a solid obtained by rotating about the x-axis a region to the left of
the curve x = f
 
(y) and to the right of the y-axis from y = c to y = d with cylindrical shells,
use the integral
V
yf y dy
c
d
= ∫ 2π ( )
More generically, you can use the formula
d
V
y
c
d
= ∫
2π (
) (
)
shell radius shell height
To find the shell radius, let y be in the interval [c, d]; the shell radius is the distance
from y to the line of rotation. To find the shell height, let f
 
(y) be the curve that bounds
the region on the right and let g(y) be the curve that bounds the region on the left; the
shell height is given by f
 
(y) – g(y). Note that the limits of integration often correspond
to points of intersection.