The Chemistry Maths Book, Second Edition

5.3 The definite integral 133

In the general case, wheny 1 = 1 f(x)is not necessarily a linear function, the integral

calculus (see Section 5.4) tells us that the area is given by the definite integral

(5.11)

whereF(x)is the function whose derivative is. The numbersF(a)

andF(b)are the values ofF(x)at the limits of integrationaand b; ais called the

lower limit, bthe upper limit, and the interval ato bis called the range of integration.

The differenceF(b) 1 − 1 F(a)in equation (5.11) is often denoted by , so that

(5.12)

It follows that in order to calculate the value of the definite integral it is normally

necessary first to evaluate the corresponding indefinite integral. For example, let

y 1 = 1 f(x) 1 = 12 x 1 + 13. The indefinite integral is

The definite integral off(x)in the rangex 1 = 1 atox 1 = 1 b(‘the integral from ato b’) is then

= 1 (b

2

1 + 13 b) 1 − 1 (a

2

1 + 13 a)

We note that the constant of integration Ccancels for a definite integral, and can

always be omitted.

EXAMPLES 5.4Definite integrals

(i)

(ii)

(iii)

ZZ

2

4

2

2

4

2

2

4

11

4

dx 1

x

xdx

x

==−

















=−











−−

−

22

1

4

1

2

1

4











=− + =

Z

2

3

2

3

2

3

33

3

3

3

2

3

19

3

xdx

x

=

















=− =

Z

1

4

1

4

222

() 23 xdxxx+=+3 434131( )( )













=+×−+×==−=28 4 24

Z

a

b

a

b

() 23 xdxxxC bbCa 3 ( )( 3 3

222

+=++













=++−+aaC+ )

ZZfxdx() =+=++=( ) 23 x dx x x C Fx 3 ()

2

Z

a

b

a

b

fxdx Fx Fb Fa() = () () ()













=−

a

b

Fx()













fx F x

dF

dx

() ()= ′ =

AfxdxFbFa

a

b

==−Z () () ()