Part II: The Answers
436
Answers
601–700
697.
8
3
π
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.
Set the functions equal to each other and solve for x in order to find the other limit
of integration:
4
4
0
4
4
0
2
2
2
2
2
x x
x
x
x
x
x
−
=
=
−
+
=
−
−
=
(
)(
)
Here, the shell height is given by 4 – (4x – x
2
) and the shell radius is x, so the integral to
find the volume is
2
4 4
2
4
4
2 2
4
3
1
4
2
0
2
2
3
0
2
2
3
4
0
2
π
π
π
x
x x
dx
x
x
x dx
x
x
x
−
−
(
)
(
)
=
−
+
(
)
=
−
+
(
)
∫
∫
= =
−
+
−
)
(
=
2 2 2
4
3
2
1
4
2
0
8
3
2
3
4
π
π
( )
( )
( ) ( )
698.
81
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