Understanding Engineering Mathematics

0

y

x

y = sin x cos x

= sin 2x

p

−p

1

2

Figure 10.10Sketch ofy=sinxcosx.

(v) y=x^4

We have already discussed this relatively simple function in Review

Question 10.1.3(v), and we know it has one minimum, at the origin.

It is symmetric about they-axis and clearly has shape similar to a

parabola – but ‘more squashed’. It is sketched in Figure 10.11.

0 x

y

y = x^4

Figure 10.11Sketch ofy=x^4.

10.2.5 Applications of integration – area under a curve

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There are many applications of integration in engineering and science, but for our purposes
they can be grouped into the categories:

• Applications in further topics in engineering mathematics such as
  Laplace Transform and Fourier series (Chapter 17) and solutions of
  differential equations (Chapter 15).


• Applications in mechanics to such things as centre of mass and moment
  of inertia.


• Geometrical applications such as calculating areas, volumes, lengths of
  curves.


• Applications in probability and statistics, such as mean values, root mean
  square values, probability in the case of a continuous random variable.

The basic principle behind applications of integration is essentially that of obtaining a
totalof a quantity by regarding it as the sum of a very large (infinite) number of elementary
quantities: