250
Worksheet 6.4.
The Method of Cylindrical Shells
1. Sketch the solid obtained by rotating the region underneath the graph of the function
over the given interval about the y- axis and find its volume.
√
a. f (x) = x, [0, 4]
y
2
x
-4
b. f (x) =
-3
√
-1
-2
1
2
4
3
x2 + 9, [0, 3]
y
-3
-2
-1
1
2
251
2. Use the Shell Method to compute the volume of the solid obtained by rotating the region
enclosed by y = 8 − x3 and y = 8 − 4x about the y-axis.
3. Sketch the solid obtained by rotating the region between the graphs of f (x) = x3 and
g(x) = 1 over [0, 1] about the vertical line x = −2.
4. Use the Shell Method to calculate the volume of the region found by rotating y = 4 − x2
on [0, 2] about the y-axis.
252
5. Use the Shell Method to find the volume of the solid formed by rotating the region below
y = 6 and above y = x2 + 2 over [0, 2] about the vertical line x = −3.
y
6
5
4
3
2
1
-8
-6
-4
-2
2
x
253
Solutions to Worksheet 6.4
1. Sketch the solid obtained by rotating the region underneath the graph of the function
over the given interval about the y axis and find its volume.
√
a. f (x) = x, [0, 4]
√
Each shell has radius x and height x, so the volume of the solid is
4
Z 4
Z 4
√
2 5/2 128
3/2
x dx = 2π
2π
=
x x dx = 2π
x
π
5
5
0
0
0
√
b. f (x) = x2 + 9, [0, 3]
√
Each shell has radius x and height x2 + 9, so the volume of the solid is
Z 18
Z 3 √
u1/2 du
2π
x x2 + 9 dx = π
9
0
18
2 3/2 =π
u
3
9
√
= 18π(2 2 − 1)
254
2. Use the Shell Method to compute the volume of the solid obtained by rotating the region
enclosed by y = 8 − x3 and y = 8 − 4x about the y-axis.
Following is a sketch of the solid.
Notice that the solid is generated by rotating two vertical strips 8 − 4x ≤ y ≤ 8 − x3
and x3 + 8 ≤ y ≤ 8 + 4x for every 0 ≤ x ≤ 2. Thus each of the two shells has radius x
and height 4x − x3 . The volume of the resulting solid is
4π
Z
2
3
0
Z
2
(4x2 − x4 ) dx
0
2
256π
4 3 1 5 x − x =
= 4π
3
5
15
0
x(4x − x ) dx = 4π
3. Sketch the solid obtained by rotating the region between the graphs of f (x) = x3 and
g(x) = 1 over [0, 1] about the vertical line x = −2.
255
Each shell has radius x − (−2) = x + 2 and height 1 − x3 , so the volume of the solid is
Z 1
Z 1
3
2π
(2 + x) 1 − x dx = 2π
(−x4 − 2x3 + x + 2) dx
0
0
= 2π
1
18π
−x5 x4 x2
−
+
+ 2x =
5
4
2
5
0
4. Use the Shell Method to calculate the volume of the region formed by rotating y = 4 − x2
on [0, 2] about the y-axis.
When the region in the figure is rotated about the y-axis, each shell has radius x and
height 4 − x2 . The volume of the resulting solid is
2π
Z
0
2
2
Z
2
(4x − x3 ) dx
0
2
x4 2
= 2π(2x − )
4 0
= 2π(8 − 4) = 8π
x(4 − x ) dx = 2π
256
5. Use the Shell Method to find the volume of the solid formed by rotating the region below
y = 6 and above y = x2 + 2 over [0, 2] about the vertical line x = −3.
Each shell has radius x − (−3) = x + 3 and height 6 − x2 + 2 = 4 − x2 , so the volume
of the solid is
Z 2
Volume = 2π
(x + 3)(6 − (x2 + 2)) dx
0
Z 2
= 2π
(4x − x3 + 12 − 3x2 ) dx
0
2
x4
2
3 = 2π 2x −
+ 12x − x 4
0
= 2π(8 − 4 + 24 − 8) = 40π