1. Chapter 13
Multiple Integrals
by Zhian Liang

2. 13.1 Double integrals over rectangles
Recall the definition of definite integrals of
functions of a single variable
Suppose f(x) is defined on a interval [a,b].

3. Taking a partition P of [a, b] into subintervals:
Using the areas of the small rectangles to approximate the
areas of the curve sided echelons

4. and summing them, we have
(1)
(2)

5. Double integral of a function of two variables defined on a closed rectangle like the following
Taking a partition of the rectangle

7. Choosing a point
in Rij and form the
double Riemann sum
(3)

8. (4) DEFINITION The double integral of f over the rectangle R is defined as
if this limit exists

9. Using Riemann sum can be approximately evaluate a double integral as in the following example.
EXAMPLE 1 Find an approximate value for the integral
by computing
the double Riemann sum with partition pines x=1 and x=3/2
and taking
to be the center of each rectangle.

10. Solution The partition is shown as above Figure
Solution The partition is shown as above Figure. The area of each subrectangle is
is the center Rij,
and f(x,y)=x-3y2. So the corresponding Riemann sum is

11. Interpretation of double integrals as volumes
(5)

12. (6) THEOREM If and f is continuous on the rectangle R, then the volume of the solid that lies above R and under the surface is

13. EXAMPLE 2
Estimate the volume of the solid that lies above the square
and below the elliptic paraboloid
Use the partition of R into four squares and choose
to be the upper right corner of .
Sketch the solid and the approximating rectangle boxes.

14. The partition and the graph of the function are
Solution
The partition and the graph of the function are
as the above. The area of each square is 1.
Approximating
the volume by the Riemann sum, we have
This is the volume of the approximating rectangular boxes shown as above.

15. The properties of the double integrals
（7）
（8）
（9） If
for all
then

16. EXERCISES 13.1
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17. 13.2 Iterated Integrals
The double integral can be obtained by evaluating two single integrals.
The steps to calculate
, where
Fix
with respect to
to calculate
Then calculate

18. (1)
(called iterated integral)
(2)
(3)
Similarly

19. EXAMPLE 1 Evaluate the iterated integrals
(See the blackboard)

20. (4) Fubini’s Theorem If is continuous on the rectangle then
More generally, this is true if we assume that
is bounded on , is discontinuous only on
a finite number of smooth curves, and the iterated
integrals exist.

21. Interpret the double integral as the volume V of the solid

22. where
is the area of a cross-section of S in the plane through x perpendicular to the x-axis.
Similarly

23. EXAMPLE 2
Evaluate the double integral
where
(See the blackboard)

24. EXAMPLE 3
,where
Evaluate

25. Solution 1 If we first integrate with respect to x,
we get

26. Solution 2 If we first integrate with respect to y, then

27. EXAMPLE 4 Find the volume of the solid S that is bounded by the elliptic paraboloid , the plane and , and three coordinate planes.

28. Solution
We first observe that S is the solid that lies under the surface
and the above the
Square
Therefore,

29. If on , then

30. EXAMPLE 5 If
, then

31. EXERCISES 13.2
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1(2), 6, 10, 16, 17,

32. 13.3 Double integrals over general regions
To integrate over general regions like
which is bounded, being enclosed in a rectangular region R .
Then we define a new function F with domain R by if
(1) if

33. If F is integrable over R , then we say f is integrable over D and we define the double integral of f over D by
(2)
where is given by Equation 1.

34. Geometric interpretation
When
The volume under f and above D equals to that under
F and above R.

35. Type I regions (3) If f is continuous on a type I region D such that
then

36. Type II regions
(4)
(5)
where D is a type II region given by Equation 4

37. it is Type I region!

38. Example 2
Find the volume of the solid that lies under the
paraboloid
and above the region D in the
-plane bounded by the line
and the parabola

40. Solution 1
Type I
Type II
Solution 2

41. Example 3 Evaluate , where D is the region bounded by the line and the parabola
D as a type I
D as a type II

42. Solution
We prefer to express D as a type II

43. Example 4 Find the volume of the tetrahedron bounded by
the planes

44. Here is wrong in the book!
Solution
Here is wrong in the book!

45. Example 5 Evaluate the iterated integral
D as a type II
D as a type I

46. Solution If we try to evaluate the integral as it stands, we
are faced with the task of first evaluating
But it is
impossible to do so in finite terms since
is not an
elementary function.(See the end of Section 7.6.) So we must
change the order of integration. This is accomplished by first
expressing the given iterated integral as a double integral.
Where
Using the alternative description of D,
we have

47. This enables us to evaluate the integral in the reverse
order:

48. Properties of double integrals
(6)
(7)
(8) if
(9) if
do not overlap except
perhaps on their boundaries like the following:

49. (10)
(11)
, the area of region D. If

50. Example 6
Solution Since
we have
and therefore
thus, using m=e-1=1/e, M=e, and A(D)=(2)2 in Property 11
we obtain

51. Exercises 13.3
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52. 13.4 DBOUBLE INTEGRALS IN POLAR COORDINATE
Suppose we want to evaluate a double integral
where is one of regions shown in the following.
(a)
(b)

53. Recall from Section 9.4 that the polar coordinates
of a point related to the rectangular coordinates
by the equations:
The regions in
the above
figure are
special cases
of a polar rectangle
Do the following partition (called polar partition)

55. The center of this subrectangle
and the area is
where

56. The typical Riemann sum is
(1)

57. If we write , then the
above Riemann sum can be written as
which is the Riemann sum of the double integral
Therefore we have

59. (2) Change to polar coordinates in a double integral If is continuous on a polar rectangle given by
where then

60. Caution: Do not forget the factor r in (2)!

61. Example 1 Evaluate , where is the region in the upper half-plane bounded by the circles
Solution
The region R can be described as

62. Example 2 Find the volume of the solid bounded
by the xy-plane and the paraboloid

63. What we have done so far can be extended to the complicated type of region shown in the following.
(3) If f is continuous on a polar
region of the form
then

64. Example 3 Use a double integral to find the area enclosed by one loop of the four-leaved rose

65. Example 4 Find the volume of the solid that lies under the paraboloid , above the plane, and inside the cylinder
Solution
The solid lies above the disk,
whose boundary circle

67. Exercises 13.4
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