Chapter 38

Some applications of

integration

74.1 Introduction

There are a number of applications of integral calculus
in engineering. The determination of areas, mean and
r.m.s. values, volumes, centroids and second moments
of area and radius of gyration are included in this
chapter.

38.2 Areas under and betweencurves

In Fig. 38.1,

total shaded area=

∫b

a

f(x)dx−

∫c

b

f(x)dx

+

∫d

c

f(x)dx

E

(^0) F
G
y
a b c d x
y 5 f(x)
Figure 38.1
Problem 1. Determine the area between the curve
y=x^3 − 2 x^2 − 8 xand thex-axis.
y=x^3 − 2 x^2 − 8 x=x(x^2 − 2 x− 8 )=x(x+ 2 )(x− 4 )
Wheny=0,x=0or(x+ 2 )=0or(x− 4 )=0, i.e.
wheny=0,x=0or−2 or 4, which means that the
curve crosses thex-axis at 0,−2, and 4. Since the curve
is a continuous function, only one other co-ordinate
value needs to be calculated before a sketch of the
curve can be produced. Whenx=1,y=−9, show-
ing that the part of the curve betweenx=0andx= 4
is negative. A sketch ofy=x^3 − 2 x^2 − 8 xis shown in
Fig. 38.2. (Another method ofsketchingFig. 38.2 would
have been to draw up a table of values.)
y
x
210
10
220
22 4
y 5 x^322 x^228 x
210 132
Figure 38.2
Shaded area

∫ 0
− 2
(x^3 − 2 x^2 − 8 x)dx−
∫ 4
0
(x^3 − 2 x^2 − 8 x)dx

[
x^4
4
−
2 x^3
3
−
8 x^2
2
] 0
− 2
−
[
x^4
4
−
2 x^3
3
−
8 x^2
2
] 4
0

(
6
2
3
)
−
(
− 42
2
3
)
= 49
1
3
square units