18.6 Calculus 619
Example 18.10 Find the derivative off(x) (x
3
10 x
2
8)
4
.We use rule 9, from Table 18.6 to solve this problem as shown. For
this problem,u(x
3
10 x
2
8).Example 18.11 Find the derivative off(x) lnx
3
10 x
2
8 .
We use rule 10, f¿(x)g¿(x)/g(x), from Table 18.6 to solve this problem as shown. For
this problem,g(x) x
3
10 x
2
8.Example 18.12 Find the derivative off(x) e
(x^3 10 x^2 8)
.
We use rule 11,f¿(x) g¿(x) #e
g(x)
, from Table 18.6 to solve this problem as shown. For
this problem,g(x) x
3
10 x
2
8.Integral Calculus
Integral calculus plays a vital role in the formulation and solution of engineering problems. To
demonstrate the role of integrals, consider the following examples.Example 18.13 Recall in Chapter 7, we discussed a property of an area known as the second moment of area.
The second moment of area, also known as the area moment of inertia, is an important prop-
erty of an area that provides information on how hard it is to bend something and therefore playsf
œ
1 x 2 13 x
2
20 x 2 e
1 x 3 10 x 2 82f
œ
1 x 2 13 x
2
20 x 2x
3
10 x
2
8f
œ
1 x 2 41 x
3
10 x
2
82
3
13 x
2
20 x 2f
œ
1 x 2 n#un (^1) #du
dx
,
A
y
y
x
■Figure 18.17
Small area element located
at a distancexfrom the
y—yaxis.
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