The Chemistry Maths Book, Second Edition

9.11 Change of variables 285

and the integral (the sum of horizontal strips) is

(9.58b)

EXAMPLE 9.30Evaluate the integral off(x,y) 1 = 111 + 12 xy

over the region R bounded by the liney 1 = 1 xand the curve

y 1 = 1 x

2

(or ), as in Figure 9.14. Calculate also the

area of R.

(i) using equation (9.58a) withg(x) 1 = 1 x

2

andh(x) 1 = 1 x,

(ii) using equation (9.58b) withp(y) 1 = 1 yand

The area of the region R is

0 Exercises 66–68

9.11 Change of variables

We saw in Section 6.3 that one of the principal general methods of evaluating

integrals is the method of substitution whereby, given a definite integral over the

variable x, a new variable of integration can be introduced by setting

xxu dx

dx

du

=,() =du

Adydxxxdx

x

=





















=−

( )

ZZ Z =

0

1

0

1

2

1

6

=+













=+







 











 





ZZ

0

1

2

0

1

xxy dy y

y

yyyydy

23

1

4

−−













=

Ixydxdy

y

=+





















ZZ

0

1

() 12

qy y() ,=

=+













=+−−





















ZZ

0

1

2

0

1

32

2

yxy dx xx x x

x

55

1

4













dx=

Ixydydx

x

=+





















ZZ

0

1

2

() 12

xy=

IfxydA fxydx

c

d

py

qy

=,= ,





















ZZZ

R

() ()

()

ddy

y

x

0

1

R

y=x

2

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Figure 9.14