The Chemistry Maths Book, Second Edition

5.3 The definite integral 135

2.If cis a third limit of integration, not necessarily between aand b:

(5.15)

This is true because the value of the integral can be written as

If clies between aand bthen the area represented by the integral on the left side of

equation (5.15) is equal to the sum of the areas represented by the integrals on the

right side.

3.When the limits are interchanged, the value of the integral changes sign:

(5.16)

This follows because

EXAMPLE 5.6Properties

(i)

.

(ii)

.

0 Exercises 29, 30

Negative areas

Consider the integral

Z

0

2

0

2

20

π

π

sinxdx=−cosx ( cos ) ( cos )π













=− −− =(()()−−−= 110

ZZ

5

2

233

2

5

2

1

3

25

117

3

xdx=− xdx

( )

=− =−

Z

2

5

233

1

3

52

117

3

xdx=−

( )

= ,

=−=

1

2

52

22

2

5

()Zxdx

Axdxxdx=+=−+−ZZ

2

3

3

5

22 22

1

2

32

1

2

()() 53

Fa Fb() ()−=− −Fb Fa() ().













ZZ

b

a

a

b

fxdx() =− fxdx()

Fb Fa Fc Fa Fb Fc() () () () () ()−= −













+−













ZZZ

a

b

a

c

c

b

fxdx fxdx fxdx() =+() ()