441
Answers
701–800
Answers and Explanations
703.
2π(e
2
+ 1)
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.
In this example, the limits of integration are given as y = 0 and y = 2, and the shell
height is e
y
– 0. Therefore, to find the volume using cylindrical shells, you use
2 0
2
π ye dy
y
∫
To evaluate this integral, use integration by parts with u = y so that du = dy, and
let dv = e
y 
dy so that v = e
y
:
2
2
2
2 2
0 1
0
2
0
2
0
2
0
2
2
2
π
π
π
π
ye dy
ye
e dy
ye e
e e
y
y
y
y
y
∫
∫
=
−
(
)
=
−
(
)
=
− − −
(
)
(
)
= =
+
(
)
2
1
2
π e
704.
π 1 1
9
−






e
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.