Double integrals, their properties and evaluations. Area. Double integral in Polar form (Lecture 8)

1. Mathematical Analysis 2

Lecture 8 - Double integrals, their properties and
evaluations. Area.
Double integral in Polar form. Substitutions in
double integrals.
Akbota Myrzakul
a.myrzakul@astanait.edu.kz
Astana IT University

2. Lecture overview

• Introduction to double integrals. Iterated integrals
• Theorem of Fubini. Properties of double integrals
• Integrals over non-rectangular regions
• Reversing the order of integration
• Simple polar regions
• Double integrals in polar coordinates
• Finding area using polar double integrals
• Changing coordinates in integrals

3. Introduction to double integrals

4. Introduction to double integrals

5. Introduction to double integrals

Definition:
Definition: The sum in the limit is called the Riemann sum.

6. Introduction to double integrals

7. Introduction to double integrals

8. Iterated integrals

Definition:

9. Iterated integrals

10. Iterated integrals

11. Theorem of Fubini

Theorem:

12. Theorem of Fubini

13. Properties of double integrals

Theorem:

14. Integrals on non-rectangular regions

Definition:

15. Integrals on non rectangular regions

16. Properties of double integrals

Theorem:

17. Properties of double integrals

18. Properties of double integrals

19. Properties of double integrals

20. Properties of double integrals

21. Reversing the order of integration

22. Reversing the order of integration

23. Racalls

24. Simple polar regions

25. Simple polar regions

26. Double integrals in polar coordinates

27. Double integrals in polar coordinates

28. Double integrals in polar coordinates

29. Double integrals in polar coordinates

30. Double integrals in polar coordinates

31. Double integrals in polar coordinates

32. Double integrals in polar coordinates

33. Finding area using polar double integrals

34. Changing coordinates in integrals

Properties
1. Let R {(r, ) | a r b, } be a polar rectangleand 0 - 2 If f is
continuous on R, then
b
f(x, y)dA f(rcos , rsin )rdrd
a
R
2. Let D {(r, ) | , h1 ( ) r h 2 ( )} be a polor region.If f is continuous on
h 2 ( )
D then f(x, y)dA
D
h 1 ( )
f(rcos , rsin )rdrd

35. Changing coordinates in integrals

36. Changing coordinates in integrals

Example : Evaluate (4y2 3x)dA where R {(x, y) | y 0,1 x 2 y 2 4}
R
Solution:
R {(x, y) | y 0,1 x 2 y 2 4} {(r, ) |1 r 2, 0 }
2
0
1
(4y 3x)dA (4(rsin ) 3rcos )rdrd
2
2
R
(15sin2 7cos )d
0
15
2

37. Changing coordinates in integrals

2. Find the volume of the solidbounded by the plane z 0
and the paraboloid z 1- x 2 - y 2
Sol :
D {(r, ) | 0 r 1, 0 2 }
V (1- x 2 - y 2 )dA
D
2
0
2
2
(1r
)rdrd
1
0

38. Changing coordinates in integrals

2. Find the volume of the solidbounded by the plane z 0
and the paraboloid z 1- x 2 - y 2
Sol :
D {(r, ) | 0 r 1, 0 2 }
V (1- x 2 - y 2 )dA
D
2
0
2
2
(1r
)rdrd
1
0

39. Changing coordinates in integrals

Example : Evaluate e-(x y )dAwhere R 2 {(x, y) | - x , - y }
2
2
R2
Consider D n {(r, ) | 0 r n, 0 2 }
Solution :
Then
-(x y )
-(x y )
e
dA
dA
lim
e
2
2
2
n
R2
lim
2
n 0
2
Dn
2
1 1 -n 2
e
rdrd
(
- e )d
lim
0
n 0
2 2
n
-r 2

40. Lecture Summary

• Introduction to double integrals
• Iterated integrals.Theorem of Fubini
• Properties of double integrals
• Integrals over non-rectangular regions
• Reversing the order of integration
• Simple polar regions
• Double integrals in polar coordinates
• Finding area using polar double integrals