Part II: The Answers
430
Answers
601–700
2
1
2
1
2
2 3 1
4
1
3
1
3
1
3
π
π
π
π
π
x x
dx
dx
x
( ) =
=
=
−
=
∫
∫
(
)
689.
8π
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.
Find the other limit of integration by determining where y = x
2
and y = 0 intersect: x
2
= 0
gives you x = 0. The shell height is x
2
– 0, so the integral becomes
2
2
2 4
2
2 0
8
2
0
2
3
0
2
4
0
2
4
π
π
π
π
π
x x dx
x dx
x
( ) =
=
=
−
(
)
=
∫
∫
690.
6
7
π
To find the volume of a solid obtained by rotating about the x-axis a region that’s 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