Page 110 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 5] INTEGRALS 101

This method of generating a volume is called the disk method because the cross sections of revolution
are circular disks. (See Fig. 5-5(a).)

Fig. 5-5

EXAMPLE. Asolid cone is generated by revolving the graph of y ¼ kx, k > 0 and 0 @ x @ b, about the x-axis.
Its volume is
3 3
b k b

ð 3 3 b
2 2
V ¼ k x dx ¼ k x ¼
0 3 0 3
Shell Method
Suppose f is a continuous function on ½a; bŠ, a A 0, satisfying the condition f ðxÞ A 0. Let R be a
plane region bound by f ðxÞ, x ¼ a, x ¼ b, and the x-axis. The volume obtained by orbiting R about the
y-axis is
ð b
2 xf ðxÞ dx
V ¼
a
This method of generating a volume is called the shell method because of the cylindrical nature of the
vertical lines of revolution. (See Fig. 5-5(b).)

EXAMPLE. If the region bounded by y ¼ kx,0 @ x @ b and x ¼ b (with the same conditions as in the previous
example) is orbited about the y-axis the volume obtained is
b b

ð 3 b 3
V ¼ 2 xðkxÞ dx ¼ 2 k x ¼ 2 k
0 3 0 3
By comparing this example with that in the section on the disk method, it is clear that for the same
plane region the disk method and the shell method produce different solids and hence different volumes.
Moment of Inertia
Moment of inertia is an important physical concept that can be studied through its idealized geo-
metric form. This form is abstracted in the following way from the physical notions of kinetic energy,
2
1
K ¼ mv , and angular velocity, v ¼ !r. (m represents mass and v signifies linear velocity). Upon
2
substituting for v
2 2
2
K ¼ m! r ¼ ðmr Þ! 2
1
1
2 2
When this form is compared to the original representation of kinetic energy, it is reasonable to
2
2
identify mr as rotational mass. It is this quantity, l ¼ mr that we call the moment of inertia.
Then in a purely geometric sense, we denote a plane region R described through continuous func-
tions f and g on ½a; bŠ, where a > 0 and f ðxÞ and gðxÞ intersect at a and b only. For simplicity, assume
gðxÞ A f ðxÞ > 0. Then