1. Chapter 15 Multiple Integrals
15.1 Double Integrals over Rectangles
15.2 Iterated Integrals
15.3 Double Integrals over General Regions
15.4 Double Integrals in polar coordinates
15.5* Applications of Double Integrals
15.6* Surface Area

2. 15.1 Double Integrals over Rectangles
Volumes and Double Integrals
A function f of two variables defind on a closed rectangle
and we suppose that
The graph of f is a surface with equation
Let S be the solid that lies above R and under the graph
of f ,that is ,

3. (See Figure 1)Find the volume of S
1) Partition:
The first step is to divide the rectangle R into subrectangles.
Each with area

4. 2) Approximation:
A thin rectangular box:
Base:
Hight:
We can approximate by

5. 3) Sum:
A double Riemann sum
4) Limit:

6. The sufficient condition of integrability:
Definition The double integral of f over the rectangle
R is
if this limit exists.
The sufficient condition of integrability:
Theorem1.
is integral on R

7. Theorem2.
and f is discontinuous only on a finite number of
smooth curves
is integral on

8. Example 1 If
evaluate the integral
Solution

9. 15.2 Iterated Integrals
Partial integration with respect to y defines a function
of x:
We integrate A with respect to x from x=a to x=b,
we get

10. The integral on the right side is called an iterated integral
and is denoted by
Thus
Similarly

11. Fubini’s theorem If f is continuous on the rectangle
then
More generally, this is true that we assume that f is bounded
on R , f is discontinuous only on a finite number of smooth
curves, and the iterated integrals exist.
The proof of Fubini’s theorem is too difficult to include
In our class.

12. If f (x,y) ≥0,then we can interpret the double integral
as the volume V of the solid S that lies above
R and under the surface z=f(x, y). So

13. Or

14. Example 1　　　　　　　
Solution 1

15. Solution 2

16. Example 2　　　　　　　
Solution

19. Example 3　　　　　　　
Solution

20. Specially If
Then

21. 15.3 Double Integrals over General Regions
Suppose that D is a bounded region which can be
enclosed in a rectangular region R. D D
A new function F with domain R:

22. If the integral of F exists over R, then we define the
double integral of f over D by

23. If the integral of F exists over R, then we define the
double integral of f over D by
A plane region D is said to be of type I if
Where and are continuous on [a,b]
Some examples of type I

24. Evaluate where D is a region of type I
A new function F with domain R:

26. If f is continuous on type I region D such that
then

27. A plane region D is said to be of type II if
Where and are continuous on [a,b]
Some examples of type II

28. If f is continuous on type II region D such that
then

29. Example 1
Evaluate ,where D is the region
bounded by the parabolas
Solution
Type I

31. Type II

32. Properties of Double Integral Suppose that functions f and g are continuous on a bounded closed region D. Property 1 The double integral of the sum (or difference) of two functions exists and is equal to the sum (or difference) of their double integrals, that is, Property 2 Property 3 where D is divided into two regions D1 and D2 and the area of D1 ∩ D2 is 0.

33. Property 4 If f(x, y) ≥0 for every (x, y) ∈D, then Property 5 If f(x, y)≤g(x, y) for every (x, y) ∈D, then Moreover, since it follows from Property 5 that hence

34. Property 6 Suppose that M and m are respectively the maximum and minimum values of function f on D, then where S is the area of D. Property 7 (The Mean Value Theorem for Double Integral) If f(x, y) is continuous on D, then there exists at least a point (ξ,η) in D such that where S is the area of D. f (ξ,η) is called the average Value of f on D

35. Example 2
Evaluate ,where D is the region
bounded by the parabolas
Solution
Type II

37. Type I

38. Example 3
Evaluate ,where D is the region
bounded by the parabolas
Solution
Type I

40. 15.4 Double Integrals in polar coordinates
A polar rectangle

41. where
The “center” of the polar subrectangle
has polar coordinates

42. The area of is

43. Change of polar coordinate in a double integral
If f is continuous on a polar rectangle R given by
, ,where , then

44. 1.If f is continuous on a polar region of the form
then

45. 2.If f is continuous on a polar region of the form
then

46. 3.If f is continuous on a polar region of the form
then

47. Evaluate ,where D is the region
bounded by the circles
Example 1
Solution 1 2 o

48. Example 2 Evaluate where D = Solution In polar coordinates, the equation x2 + y2 –2x = 0 becomes The disc D is given by D’: Therefore