082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

15 Applicationsof
integration
Contents 15.1 Introduction 471
15.2 Averagevalueofafunction 471
15.3 Rootmeansquarevalueofafunction 475
Reviewexercises15 479
15.1 INTRODUCTION
Currents and voltages often vary with time. Engineers may wish to know the average
valueofsuchacurrentorvoltageoversomeparticulartimeinterval.Theaveragevalue
of a time-varying function is defined in terms of an integral. An associated quantity is
therootmeansquare(r.m.s.)valueofafunction.Ther.m.s.valueofacurrentisusedin
the calculation of the power dissipated by a resistor.
15.2 AVERAGEVALUEOFAFUNCTION
Suppose f (t)isafunction defined ona t b. The area,A, under f isgiven by
A =
∫ b
a
f dt
A rectangle with base spanning the interval [a,b] and heighthhas an area ofh(b −a).
Supposetheheight,h,ischosensothattheareaunder f andtheareaoftherectangleare
equal. This means
h(b−a)=
h =
∫ b
a
f dt
∫ b
a f dt
b −a
Thenhiscalledtheaverage valueofthefunctionacrosstheinterval[a,b] andisillustrated
inFigure 15.1:
average value =
∫ b
a f dt
b −a