Part II: The Answers
444
Answers
701–800
706.
4π(2 ln 3 – 1)
The region is being rotated about the line x = 4, which is parallel to the y-axis. To find the
volume of a solid obtained by rotating about the y-axis a region that’s under the curve
y = f
 
(x) and above the x-axis from x = a to x = b with cylindrical shells, use the integral
V
xf x dx
a
b
= ∫ 2π ( )
More generically, you can use the formula
d
V
x
a
b
= ∫
2π (
) (
)
shell radius shell height
To find the shell radius, let x be in the interval [a, b]; the shell radius is the distance
from x to the line of rotation. To find the shell height, let f
 
(x) be the function that
bounds the region above and let g(x) be the function that bounds the region below; the
shell height is given by f
 
(x) – g(x). Note that the limits of integration often correspond
to points of intersection.
Because the region being rotated is to the left of the line of rotation x = 4, the shell
radius is (4 – x). To find the volume using cylindrical shells, use the following integral:
2
4
1
2
4 1
2 4
2 4 3 3 4 1
1
3
1
3
1
3
π
π
π
π
(
)
ln
ln
( ln
− ( ) =
−
( )
=
−
(
)
=
− −
∫
∫
x x
dx
x
dx
x x
− −
(
)
=
−
=
−
1
2 4 3 2
4 2 3 1
)
( ln
)
( ln
)
π
π
707.
10
21
π
The region is being rotated about the line y = 1, which is parallel to the x-axis. 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.