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

13 Integration
Contents 13.1 Introduction 428
13.2 Elementaryintegration 429
13.3 Definiteandindefiniteintegrals 442
Reviewexercises13 453
13.1 INTRODUCTION
When a function, f (x), is known we can differentiate it to obtain the derivative, df
dx .
Thereverseprocessistoobtainthefunction f (x)fromknowledgeofitsderivative.This
process iscalledintegration. Thus,integration isthe reverse of differentiation.
A problem related to integration is to find the area between a curve and thexaxis.
At first sight it may not be clear that the calculation of area is connected to integration.
This chapter aims to explain the connection. An area can have various interpretations.
For example, the area under a graph of power used by a motor plotted against time
representsthetotalenergyusedbythemotorinaparticulartimeperiod.Theareaunder
a graph of current flow into a capacitor against time represents the total charge stored
by the capacitor.
Circuits to carry out integration are used extensively in electronics. For example, a
circuittodisplaythetotaldistancetravelledbyacarmayhaveaspeedsignalasinputand
may integrate this signal to give the distance travelled as output. Integrator circuits are
widely used in analogue computers. These computers can be used to model a physical
system and observe its response to a range of inputs. The advantage of this approach is
thatthesystemparameterscanbevariedinordertoseewhateffecttheyhaveonsystem
performance.Thisavoidstheneedtobuildtheactualsystemandallowsdesignideasto
be explored relatively quickly and cheaply by an engineer.