Understanding Engineering Mathematics

9.4 Applications

We will be devoting a lot of Chapter 10 to applications of integration, and you will have
ample opportunity to see applications of integration later in the book. For now we will
apply the skills learnt in this chapter to look ahead to Laplace transforms and Fourier
series, covered in Chapter?.

1.Supposef(t)is a function oft, defined fort≥0. Then the integral

f(s) ̃ =

∫∞

0

f(t)e−stdt

is called theLaplace transform off.t/(the use oftas the variable is conventional,

since the Laplace transform is usually applied to functions of time). We will study

this in detail in Chapter 17. We will want to evaluate this integral for all the simple

elementary functions. In order to do this we will need to integrate finite integrals of

the sort

∫a

0

f(t)e−stdt

for a constanta. Do this for the functionsf(t)=1,t,t^2 ,et,sint,cost. Your results

should be functions ofaands. If you feel confident enough take the limit as a tends

to infinity, thereby obtaining the laplace transforms of the functions – see Chapter 17

for the results.

2.Letf(t)be a function with period 2π. Then under certain conditions it can be expanded
in a series of sines and cosines in the form

f(t)=

a 0

2

+

∑∞

n= 1

ancosnt+

∑∞

n= 1

bnsinnt

This is called aFourier (series) expansionforf(t). We study such series in detail in

Chapter 17, here we want to anticipate some of the results used there. The key task is

to determine the coefficientsan,bnof the Fourier series for a given functionf(t).This

involves the use of a number of integrals of products of sines and cosines, which we

ask you to evaluate here.

Ifm,nare any integers, then:

∫π

−π

sinmtsinntdt=0ifm   =n

∫π

−π

cosmtcosntdt=0ifm   =n

∫π

−π

cos^2 ntdt=π

∫π

−π

sin^2 ntdt=π

∫π

−π

sinmtcosntdt=0forallm,n

∫π

−π

sinmtdt=

∫π

−π

cosmtdt= 0