The Chemistry Maths Book, Second Edition

172 Chapter 6Methods of integration

Definite integrals

When the variable of integration of a definite integral is changed, fromxtou(x)say,

the limits of integration must also be changed. If the range of integration over xis

from ato bthen the range of integration overu(x)is fromu(a)tou(b):

(6.11)

EXAMPLE 6.9Integrate.

Substitute u 1 = 1 cos 1 x and du 1 = 1 −sin 1 x 1 dx. When x 1 = 10 , u 1 = 1 cos 101 = 1 + 1 ; when x 1 = 1 π,

u 1 = 1 cos 1 π 1 = 1 − 1. Therefore

since interchanging the limits changes the sign of the integral. Then

EXAMPLE 6.10Find the area of the circle whose equation isx

2

1 + 1 y

2

1 = 1 a

2

.

As in Example 5.11, let Abe the area of that quarter of the circle that lies in the first

quadrant, in which both xand yare positive (Figure 5.12). Then and

As in Example 6.8, the integral is evaluated by means of the substitutionx 1 = 1 a 1 sin 1 θ,

and the new integration limits are θ 1 = 10 when x 1 = 10 and θ 1 = 11 π 22 when x 1 = 1 a.

Therefore,

The area of the circle is four times this.

0 Exercises 33–39

Aa d

aa

==+













=

2

0

2

2

2

0

2

24

Z

π

π

π

cosθθ θsin cosθ θ

2

Aaxdx

a

=−Z

0

22

yax=−

22

ZZ

0

2

1

1

2

1

1

3

3

2

3

π

cos sinxxdx udu

u

==

















=

−

+

−

+

Iuduudu=− =+

+

−

−

+

ZZ

1

1

2

1

1

2

Ixxdx=Z

0

π

cos

2

sin

ZZ

a

b

ua

ub

f x dx() g u du()

()

()

=