Higher Engineering Mathematics

SOME APPLICATIONS OF INTEGRATION 377

H

Now try the following exercise.

Exercise 149 Further problems on mean

and r.m.s. values

1. The vertical heighthkm of a missile varies
   with the horizontal distance dkm, and is
   given byh= 4 d−d^2. Determine the mean
   height of the missile fromd=0tod=4 km.
   [2^23 km].


2. The distances of pointsyfrom the mean value
   of a frequency distribution are related to the

variatexby the equationy=x+

1

x

. Deter-
mine the standard deviation (i.e. the r.m.s.
value), correct to 4 significant figures for
values ofxfrom 1 to 2. [2.198]
3. A currenti=25 sin 100πtmA flows in an
electrical circuit. Determine, using integral
calculus, its mean and r.m.s. values each cor-
rect to 2 decimal places over the ranget= 0
tot=10 ms. [15.92 mA, 17.68 mA]
4. A wave is defined by the equation:
v=E 1 sinωt+E 3 sin 3ωt
whereE 1 ,E 3 andωare constants.
Determine the r.m.s. value ofvover the

interval 0≤t≤

π

ω

.

⎡

⎣

√

E 12 +E^23

2

⎤

⎦

38.4 Volumes of solids of revolution

With reference to Fig. 38.6, the volume of revolution,
V, obtained by rotating areaAthrough one revolution
about thex-axis is given by:

V=

∫b

a

πy^2 dx

y

0 x = ax = bx

y = f(x)

A

Figure 38.6

If a curvex=f(y) is rotated 360◦about they-axis

between the limitsy=candy=dthen the volume

generated,V, is given by:

V=

∫d

c

πx^2 dy.

Problem 5. The curvey=x^2 +4 is rotated one

revolution about thex-axis between the limits

x=1 andx=4. Determine the volume of solid

of revolution produced.

Revolving the shaded area shown in Fig. 38.7, 360◦

about thex-axis produces a solid of revolution

given by:

Volume=

∫ 4

1

πy^2 dx=

∫ 4

1

π(x^2 +4)^2 dx

=

∫ 4

1

π(x^4 + 8 x^2 +16) dx

=π

[

x^5

5

+

8 x^3

3

+ 16 x

] 4

1

=π[(204. 8 + 170. 67 +64)

−(0. 2 + 2. 67 +16)]

=420.6πcubic units

Figure 38.7

Problem 6. Determine the area enclosed by

the two curvesy=x^2 andy^2 = 8 x. If this area

is rotated 360◦about thex-axis determine the

volume of the solid of revolution produced.