The Chemistry Maths Book, Second Edition

9.12 Exercises 293

63.Evaluate on the circle with parametric equationsx 1 = 1 cos 1 θ,

y 1 = 1 sin 1 θ, (i)from A(1, 0) to B( 0, 1) and (ii)around a complete circle (θ 11 = 101 → 12 π).

(iii)Confirm that the differential 2 xy dx 1 + 1 (x

2

1 − 1 y

2

) dyis exact.

Sections 9.9

64.Evaluate the integral and show that the result is independent of

the order of integration.

65.Evaluate the integral

Sections 9.10

Evaluate the integral and sketch the region of integration:

66. 
    
    
    
    67. 
        
    
    
    
    


67. (i)Show from a sketch of the region of integration that

,

(ii)evaluate the integral.

Section 9.11

Transform to polar coordinates and evaluate:

69. 
    
    
    
    70. 
        
    
    
    
    


70. ZZ

0

∞

0

∞

e x dx dy

−+()xy

22

2

ZZ

−−

−+

• 
  

∞

∞

∞

∞

exydxdy

222 xy 3

2212

()

ZZ ()

0

1

0

1

2

2

2

−

• 
  

x

()xxydydx

ZZ ZZ Z Z

0

2

24

22

3

0

1

0

22

3

4

0

y

yx

x dx dy x dy dx

−

−−

−

=+

()/

00

42

3

()/x

xdydx

+

ZZ

00

2

22

a ax

xy dydx

−

ZZ

0

22

22

x

x

()x y dyd+ x

ZZ

00

2

π

R

r

edrd

−

cos sinθθ θ.

ZZ

0

3

1

2

22

()x y xy dxdy+

Z

C

2

22

xy dx x y dy+−













()