Multiple integrals

1
Multiple integrals

• BET2533 Eng. Math. III
• R. M. Taufika R. Ismail
• FKEE, UMP

2
Introduction

• The integration of multivariable functions are
  denoted as

Double integrals of two variables functions
Triple integrals of three variables functions
3
Double Integrals

• The graph of function of two variables f(x,y) is
  surface, plotted in 3-D. The integral of f(x,y)
  over the region R is the volume between the graph
  and the region R.

4
If f(x,y) 1, then the numerical value of the
volume is equal to value of the area of the
region R. Thus,
5
Iterated integral

• If f(x,y) is continuous in a rectangular region
• then

6

• Usually the bracket are not shown, and these are
  done with the understanding that the inner
  integrations should be performed first
• We also deal with the inner integrations with
  variables limits

7
Example 1

• Evaluate the iterated integrals.
• (i)
• (ii)

8
Solution

• (i) Using the definition of iterated integrals,
  we obtain

?
9

• (ii) Using the definition of iterated integrals,
  we obtain

?
10
Double integrals Fubinis theorem

• According to Gaudio Fubini (1879-1943), the
  double integrals of any continuous function can
  be calculated as an iterated integrals Fubinis
  theorem

11

• Fubinis theorem on a rectangular region
• if
• then

R
12

• Fubinis theorem on any region

Type I
Type II
13
Example 2

• Evaluate
• (i)
• (ii)
• (iii)

14
Solution

• (i) The region R is illustrated as

R
15

• Hence

?
16

• (ii) The region R is illustrated as

R
?
17

• (iii) The region R is illustrated as

R
?
18
Example 3

• Find the volume of the solid bounded by

19
Solution
The solid is described as
R
20
The base of the solid is the region R
R
21
Thus, the volume V is given by
22
Now let .
Hence,
?
23
Example 4

• Evaluate
• (i)
• (ii)

24
Solution

• (i) The region R is illustrated as

R
25

• But the inner integral cannot be
  integrated.
• However, we can reverse the order of integration
  dydx to dxdy. Then the region become

R
26

• Hence

?
27

• (ii) The region R is illustrated as

R
28

• But the inner integral cannot be
  integrated.
• However, we can reverse the order of integration
  dxdy to dydx. Then the region become

R
29

• Hence

?
30
Review Polar coordinates
Polar coordinates
y
r
?
x
31
Cylindrical polar coordinates
32
Spherical polar coordinates
33
Double integrals in polar coordinates

• When involving circular shape, it is easier to
  evaluate in the polar coordinates
• Suppose that we can convert a function z(x,y) to
  z f(r,?) then we may evaluate the integral of
  f(r,?) in the polar coordinates

34
z f (r,?)
V
dV
z f (r,?)
R
dA
A
35
y
dr
dA
r
rd?
d?
?
x
36
Example 5

• Evaluate the following integrals by changing to
  polar coordinates.
• (i)
• (ii)

37
Solution
(i) The region of integration R is described as
38

• Therefore

?
39
(ii) The region of integration R is described as
R
40
Therefore
?
41

• Therefore

42
Example 6

• Use the polar coordinates to solve Example 4.

43
Solution
The solid and its base is described as
R
R
44
Thus, the volume V is given by
?
45
Triple integrals Cartesian coordinates

• Triple integrals in Cartesian coordinates is an
  integration of a function of three variables
  f(x,y,z) on a 3-D closed region G

46
(No Transcript)
47
G
R
48
Example 7

• If G is the region in the first octant bounded
  by y x2, z y 1, xy-plane and yz-plane,
  evaluate
• (i) where
• (ii) The volume of the region G

49
Solution

• The region G and its projection R on the
  xy-plane is as shown below

50

• (i)In this case we have f(x,y,z) 6z, z1(x,y)
  0 and z2(x,y) 1 y. The projection of G on the
  xy-plane is a region R bounded by x 0, x 1, y
  x2 and y 1. Thus we obtain

51
?
52

• (ii)

?
53
Triple integrals Cylindrical coordinates
54
Example 8

• Use cylindrical coordinates to find the volume
  of the solid bounded by
• and
  .

55
Solution

• The solid of the region G and its projection R
  on the xy-plane are

56

• Using relation x2 y2 r2, we obtain
• z2 v(25 r2). We also have z1 0. Thus,

?
57
Example 9

• Use cylindrical coordinates to find the volume
  of the solid bounded by the paraboloid
  and the plane .

58
Solution

• The solid of the region G and its projection R
  on the xy-plane are

z
y
3
G
R
x
3
-3
3
y
R
3
-3
x
59

• Thus, the required volume is

?
60
Triple integrals Spherical coordinates
z
? sin f d?
? sin f
d?
d?
d?
?
f
? df
?
df
y
?
x
61
? sin f d?
d?
? df
62
Example 10

• Find the volume of the solid that lies above the
  cone and inside the sphere .
  

63
Solution

• The solid of the region G as illustrated below

64

• Thus, the required volume is

?