Multiple Integrals

1
Multiple Integrals
Chapter 12

• 170 121 Engineering Mathematics II
• 5 ??????? 2546

2
Integrals
Integration ???????????????? ??????????????
differentiation ??????? ??????????????????????
???????? ??????? ?????????????????????????????????
???? ????????????????????????????
????????????????????????????????
f(x)
f(x)
?????????????? dx ?????????????????????????????? ?
?? x ???? f(x) ??????????????????? ???????
f(x)dx ????????? 1 ????????? ????????????????? ??
??????????????? ?????????????????? f(x)dx
???????????????????????????????????????????? ???
??????????????????????????? ???????????????? ?????
??????????????????????????????
x
dx
3
Multiple Integrals
Multiple integration ????????????????????????
???????????????????? ?????????????????????????
????????????????????????????????????????
???????????????????????????????????????????? -
????????????????????????????????
?????????????????????????????? ???????????????????
?????????????????? ????????????????????????????
???????????????????????? ?????????????????????????
????????????????????????????? ????????????????????
????????????

4
Double Integrals over Rectangles
Multiple integral ???????????????????????????
???????????????????????????????? ?????????????????
??????????? Double integral ????????Double
integral ??????????
???????????
????????????
?????????????????????????
??????
????????? R ???????????????????????
???????????????? Rieman sum
d
c
????????
?????
a
b
5
Geometrical Interpretation of Double Integrals
??????????????????????????????????????????????
?????????? ????????????????? ??????? 1 ????
????????? Double integral ?????????????????
function f(x,y) ???????????? ???? Double
integral ???????????????????????????????????????
(????????????????????????????)
z
???????????????????????????? ?????????????????????
?????? ????? Double integral
????? XY ???????????
x
y
6
Geometrical Interpretation of Double Integrals
(cont.)
???????????? Slide ??????? 2 ?????????????????????
???????????????????? ?????????????????????????????
???????????
?????????????????????????????
???????????????????????????
z
??????????????????????????????
??????????????????????????????????????
????????????????????? ????????????????
x
y
7
Properties of Double Integrals
1.
2.
3.
4.
5.
If
R1
R2
R
8
Fubinis Theorem for Calculating Double Integrals
??????????????????????????????????????????????????
? ???????????????? (??????) ???????? z 4 - x -
y ???????????????????????????????
???????????????? Double intregral ??????
???????
???????????????? ????????????
?????? x ????????
????????
??????????????
?????????????????????
9
Examples Fubinis Theorem
??????????????? ??????????????????????????????
????????????????????????????? x ????
????????
???????
???????????????? ????????????
???????????? y ???????? ?????
??????????????
?????????????????????
????????????????????? ????????????????? Fubini
????
10
??????????? Double Integrals
?????????????????????????????????????????????
??????????? 2 ?????? x ??? y ??????? ?????????????
????????????????????????????????
1. ?????
???????????????????? (y) ????????? x
???????????? ??????????????????? ?????????
function ??? x ?????????? (????????? y
????????????? c ??? d ??????)
2. ????????????????????????????????1
?????????????????????? (????????? x)
???????????? ?????????????????? f(x,y) ???????? x
???? ???????????????? ???????? y ???????????
??????????? Fubini ????????????????????
11
Double Integrals over Bounded Nonrectangular
Regions
??????????????????????????????????????????????????
??? ?????? ????????????? Double Integration
??????????????????????????????????????????
??????????? Rieman
????????????????????????????????????
???????????????? ????????????????? ?????
??? ???? ??????????????????????????? ??????????
??????????????? ???????? dA ??????????????????????
???? f(x,y) ????? ???????????????????? XY ???????
12
Double Integrals over Bounded Nonrectangular
Regions
?????????????????? Double integral
??????????????????????????????? ????????????????
???????????????????? ?????????????? ???? f(x,y)
?????
??????????????????? ?????? ????? Double integral
???????????????????????????? ?????????????????????
??????????
13
Fubinis Theorem
???????? Fubini ??????????? Double integral
????????????????????? ????????????? ????????
??? f(x,y) ??????????? ??????? (region) R
1. ??? Region R ???????????????
??? ??????????????? a,b ?????
1. ??? Region R ???????????????
??? ??????????????? c,d ?????
14
Double Integral in dy dx Order
????????? 1 Region R
????????? ??? y ????????????? g1(x) ??? g2(x)
???????????????????????????????? y ????
???????????? y ????? ???????????????????????
??????????????????????????? x
1. ?????
???????
2. ?????? (?????????? y)
?????????????? H(x) ??????????????????? x
???????????
3. ???????????????? 2 ???????? x
15
Example Double Integral in dy dx Order
?????????????????????????????????????????????
z 3- x - y ????????????????? ????????????????
??????
1. ????????????????????
1.1 ?????????????? y ?????? ?????????????????????
???? ???? ?????????????????? y 0 ????????????
y x
????? z 3-x-y
1.2 ?????????????? x ?????? ???????????? x
???????? x 0 ???????? x 1
???????????????????????
????????? ???????
???????????????????? ?????????????????
16
Example Double Integral in dy dx Order
(continued)
???????
????? z 3-x-y
2. ??????????????????
???????????????????????
3. ?????????????????????
???? y 0
17
Procedure for Finding Limit of Integration in dy
dx Order
???????????????? Multiple integral
??????????????????????????????????????????????????
????????????????????????????????????????
????????????????????????????????
???????? ???????? ??????????????????????
?????????????????????????????? x2y2
1 ?????????? xy 1 ????????
1. ???????????? ?????????????????
?????????????????????? ???????????????????????????
????? y ????
2. ????????????????????????????? y ????
????????????????????????????????????? ????????????
?????????? x (???????????????? y g1(x)
????????????? y g2(x))
????????????? ???????????
?????????
3. ??????????? x ????????????????????????????????
x
?????????????
????????? x 0 ????????? x 1
???????????????????
18
Double Integral in dx dy Order
????????? 2 Region R
????????? ??? x ????????????? h1(y) ??? h2(y)
???????????????????????????????? x ????
???????????? x ????? ???????????????????????
??????????????????????????? y
1. ?????
???????
2. ?????? (?????????? x)
?????????????? G(y) ??????????????????? y
???????????
3. ???????????????? 2 ???????? y
19
Example Double Integral in dx dy Order
?????????????????????????????????????????????
z 3- x - y ????????????????? ????????????????
??????
1. ????????????????????
1.1 ?????????????? x ?????? ?????????????????????
???? ???? ?????????????????? x y ?????????????
x 1
1.2 ?????????????? y ?????? ???????????? y
???????? y 0 ???????? y 1
????????? ???????
y
1
?????????? ???? x 1
??????????? ???? x y
x
0
1
20
Example Double Integral in dx dy Order
(continued)
???????
2. ??????????????????
3. ?????????????????????
y
x
0
1
21
Procedure for Finding Limit of Integration in dx
dy Order
?????????????????????????????????? x ????
???????????????????
???????? ?????????????????????????
?????????????????????? x2y2 1?????????? xy 1
1. ????????????????????????????? x ????
?????????????????????????????????????? ???????????
??????????? y (???????????????? x h1(y)
?????????????? x h2(y))
????????????? ???????????
??????????
2. ??????????? y ????????????????????????????????
y
????????? y 0 ????????? y 1
???????????????????
22
Example Double Integral
??????????
??? R ???? Region ????????
??????
1. ?????????????????????? ???????????????? y ????
2. ???????????????????????
3. ?????????
4. ????????????????
5. ???????????????????
23
Example Double Integral (continued)
??????????
??? R ???? Region ????????
???????????????????????????? x ????
1. ???????????????????????
2. ?????????
3. ????????????????
??????????????????????????????
4. ???????????????????????? ..
??????????????? ??????????????????????????????
???????????????? ??????????????????????? ?????????
??????????????????
24
Example 2 Double Integral
???????
??? R ???? Region ??????????????
??? ????????
??????
??????
1. ?????????????????????? ???????????
??????????? ????????? y ???? ????????????????????
1.1 ?????????????
1.2 ???????????????
??????????? x
??????????????????? ????????????????? (0,1) ???
(3,4) ??????????????????? x ??? 0 ???????????????
3
???????
25
Example 2 Double Integral (continued)
2. ????????????????
3. ???????????????????
26
Example 3 Double Integral
???????
??? R ???? Region ??????????????
??? ????????
??????
???????????????????????????? x ????
???????????????????? x ???? ??????????????????????
?? 2 ??????????
(3,4)
??????? R1
???????????????
??????????????
????????? y ???
R2
??????? R2
(0,1)
???????????????
R1
??????????????
????????? y ???
???????
???
27
Example 3 Double Integral (continued)
2. ????????????????
??????? R1
??????? R2
3. ???????????????????
28
Areas of Bounded Regions in the Plane
??????????????? f(x,y) 1 ?????????? Double
integrals ?????????????????????????? R ?????
???????? ??????? R ?????????????????????
???????????
????? Sn ????????????????????????? R
????????????????????????????????????
???????????????? ????????????????? ?????
???????????????????????????????
29
Example 1 Areas of Bounded Regions in the Plane
?????????????? R ?????
?????? 1. ????????????????????
??????????????? ?????????????
???????????????
??????? x ???????????
???
Area
30
Example 2 Areas of Bounded Regions in the Plane
???????????????? R ?????????????????????? y x2
??? ??????? y x2
??????
1. ??????????????????????? ???
???????
???????????????????????? ???
???????? ???
????????? (-1,1) ??? (2,4)
2. ????????????????????
??????????????? ?????????????
???????????????
??????? x ???????????
3. ?????
Area
31
Example 2 Areas of Bounded Regions in the Plane
(cont.)
???????????????????????????????????? dx dy
?????????????????
y
??????????????????????????????????????????????????
?????? ????????????????? ?????????????????????????
??? 2 ????
(2,4)
?????? R1 ??????????
R2
(-1,1)
x
R1
?????? R2 ??????????
?????????????
32
Average Value
????????????????? f(x,y) ?? Domain R
????????????????
???????? ???????????????? f(x,y) xcos(xy)
???????? R
??????
1. ?????
2. ??????????????
33
Moments and Centers of mass
?????????????????????????????? Double integrals
???????? Moment ?????????????????? ???????????????
???????????????
??? ?????????????????????????????
? (??????????? ?????????????????)
??????????????????????????
First order moments
??????????? x
??????????? y
Center of Mass (???????????????)
???
????????????
34
Moments and Centers of mass (continued)
Moment of inertia (second order moment)
??????????? x
??????????? y
??????????? (0,0)
Moment of inertia about a line L
??? r(x,y) ???????????????? (x,y) ?????????? L
Radii of Gyration (???????????????)
???????????????? x
???????????????? y
???????????????? (0,0)
35
Example Moments and Centers of mass
????????????????????????????????????
?????????????????????? (x,y) ???? 6x6y6 kg/m2
???
First order moment
Center of Mass
36
Example Moments and Centers of mass (continued)
Moment of inertia
Radii of Gyration
37
Centroids of Geometric Figures
???????????????????????????????????????????????
???????????????????????????????
Centroids ???????????????????
???????? ??????? Centroid ??? Region R
??????????????? ???????? y x2 ?????????? y x
???
38
Polar Coordinate System
Polar coordinate system ??????????????????????????
??? 2 ????????????????? ???????????? x
(??????????????????????????? ?????????????????????
?????????????) ??????????? ???????????????????
(0,0) ????????????????
??? (1,1) ?????? Rectangular coordinate system
?????????? ???????? Polar coordinate system
?????????
????????
y
(1,1)
??????
?????? Polar coordinate
x
?????????????????????? Rectangular ???Polar
coordinate
39
Polar Coordinate System
??????????? Polar coordinate ?????????????????????
??????? ???????????????????
Reflection of a point about the x-axis
?? Rectangular coordinate ??????? (x,y) ???? ???
(x,-y) ??????? ?????? ?????????????? x
?? Polar coordinate ??????? (r,q) ???? ??? (r,-q)
??????? ?????? ?????????????? x
Reflection of a point about the y-axis
?? Rectangular coordinate ??????? (x,y) ???? ???
(-x,y) ??????? ?????? ?????????????? y
?? Polar coordinate ??????? (r,q) ???? ???
(r,p-q) ????(-r,-q) ??????? ??????
?????????????? y
40
Polar Coordinate System
Reflection of a point about the origin
?? Rectangular coordinate ??????? (x,y) ???? ???
(-x,-y) ??????? ?????? ?????????????? Origin
?? Polar coordinate ??????? (r,q) ???? ??? (-r,q)
??????? (r,pq) ??????? ?????? ??????????????
Origin
????????????????? Polar coordinate
??????? q c ???????????????????????????????????
(0,0) ?????????????? x ??????? c ???????? r c
???????????????????????? c ????????????????? ???
(0,0)
Polar coordinate ?????????????????????????????????
??? ????????????????????????????????? ????????????
???? Rectangular coordinate
41
Polar Graphs
Polar graph ?????????????? r function of q
??????????????????????????????? ??????????????????
??????
1. ???????????? q 2. ????? r f(q)
????????????????? 3. Plot ??? (r,q) ??? ??????
q ?????????? ?????? (0,0) ??????? r
???? r 1 - cos(q)
42
Advantage of Polar Coordinate System
???? Rectangular coordinate (x,y)
?????????????? ???????????????????????????????????
?????????????????????????????? ??????????????????
???????????
B
A
C
????????? Rectangular coordinate
???????????????????? ??????? 3 ??????????
-1
1
2
-2
(0,0)
R2 2
???? A
R1 1
???? B
???? C
????????????? Polar coordinate ???????????????????
??????????????
????????????????????? Rectangular coordinate ???
43
Double Integrals in Polar Form
??????? Double integral ?????
??????? Rectangular Coordinate
???? ???????????? R ??????????????????????????
??????? ???????
?????
????? Double integral ?????? Polar
coordinate ?????????????? ????????????? R
????????????????????????????????????
????????? Take limit ???
?????
44
Double Integrals in Polar Form
????????????????????? Double integral ??????
Rectangular coordinate ?????????? Polar
coordinate ??? DA ???? dA ???????????????????
?????? Polar coordinate DA ??????????? ??????????
Large sector - ?????????? small sector
?????????????????????????????? Polar coordinate
????????
45
Finding Limits of Integration in Polar Form
?????????????????????????????????? Polar
coordinate ??????????????? ????????? r
???????????? function ??? q ??????????????? q
???????????? ??????????????
1. ??????????????????? r ???? ????????? ???
?????????? ???
?????????
??????????
2. ????????????????? q
???????????
??????????
???????????????????????
46
Finding Limits of Integration in Polar Form
?????????????????????????????????????? Polar
coordinate ??????????
1. ??????????? r ??? ?????????? ????????????
??????????????????????????? Region
????????????????? ?????? Rectangular coordinate
????????????????????? ???????????? Polar
coordinate ???? ??????????
?????????? Polar form ???
?????????? ???????
???
???? r 2
?????????? Polar form ???
????????? ???????
????
47
Finding Limits of Integration in Polar Form
(continued)
???????????????????? Region ????????????
?????????????????? ???????? ????????????? (0,0)
??????????? - ??????????????? (0,0)
?????????????????????????????????? -
?????????????? (0,0) ?????????????????????????????
??????
????????????????? ???????????????????????
?????????????????? ??????????????????????
48
Finding Limits of Integration in Polar Form
(continued)
2. ????????????????????????????? q
???????????????????????? ?????????????????????????
????????????????????? Region
???????????????????????????
(1,1)
?????????????????????????
3. ?????????????????????????????????????????
????????????????
??????????????? r ?????? !
49
Example Finding Limits of Integration in Polar
Form
??????????????????????????????? f(r,q) rcosq
????? Polar form ??? Region ?????
1. ??????????? r ???????????????? Region R
????????? r1 ?????????????? r2 ????????????
r1 ????????????? r2 ??????????????
2. ?????????????? q ????????????????? ????????????
????????????????????????
??????????
??????????????????????
3. ???????
R
?????????
50
Area in Polar Coordinates
???????????????????? f(r,q) 1
????????????????????????????
?????????????????????
Area
????????
???????????????????
??????????????????????????????? 4 ????
?????????????????????????????????????? 4
1. ??????????? r ?????????????????????????????????
? ??????????????????? 0 ?????????????????
2. ??????????? q
?????????????????????? q 0
?????????????? ????????????????? ??????? (0,0)
???????????? r 0
???
??????
????
????
51
Area in Polar Coordinates (continued)
????????????????????????????????
????????????????????????
???????????????????????????
?????????????????
52
Changing Cartesian Integrals into Polar Integrals
?????????????? Rectangular coordinate ???? Polar
coordinate ????????????????? ?????????????????????
??????????????????
???????? ??????????
???????? ???????????????????????????? 1/4
??????????????
??????????????????????????????
????????????????
???????????????????
??????????????
53
Changing Polar Integrals into Cartesian Integrals
??????????????????????????????? Polar coordinate
?????????????????????
1. ?? function f(x,y) ???????????? x ????
rcosq ?????? y ???? rsinq ????? f(r,q) ????
2. ????????????????????????????? r ??? q
??? ????????? r 0 ?????????? r 1
????????? q 0 ????????? q p/2
3. ?????????????????????
4. ?????
54
Example Changing from rectangular to polar form
??????????
(?????????????????????????????? ????? XY ???????)
?????? 1. ?????????????? Polar form ???
??? ????????? r 0 ?????????? r
????????? q 0 ????????? q 2p
2. ????????????????????????? r ??? q
3. ?????????????????????
4. ?????
55
Triple Integrals in Rectangular Coordinates
????????????????????? function 3
???????????? Rectangular coordinate
??????????????????
?????????????????????????????? Region D
?????? ?????????? D ??????????????????????????????
? ????????????? Dx ??????????? Dy ??????????? Dz
?????????????????????
????????????? Rieman sum ????
Dz
Dv
Dx
Dy
???????? Dv ???????? 0 ?????
56
Properties of Triple Integrals
1.
2.
3.
4.
5.
If
D1
D
D2
57
Comparison Between Double integrals and Triple
integrals
?????? Double integral ??????????????????????
? dA (??? A ???????? Area) ??????????????????
???????? (R) ?????????????????? 2 ???? (????? XY)
z
????????? Triple integral ???????????????????? dV
(??? V ???????? Volume) ??????????????????????
???? (D) ????????????? 3 ????
D
dV
y
???????? ??????????????????????? 3 ???
?????????????????????????????? ??????? ???????????
??????????????? 3 ????? ?????????????? Triple
integral
x
58
Procedure for Finding Limits of Integration in
Triple Integrals
?????????????????????????????? Triple
integral ?????????????????????????????
????????? ????? ?????????????????? D ????????????
3 ??????????
?????????????????????????????
1.?????????????????????? ???? ???????????? ???????
? z, y, ??? x ???????? ????? ?????????????????????
dz dy dx
????????
2. ??????????????????????????????????????????????
(????????? ??? z) ???????????????????????????????
????????? (???????????? x ??? y)
??????????? ????????????????????? z
f2(x,y) ??????????????????????? z f1(x,y)
??????????
?????????????????????????????????????????????????
???????????????? z f2(x,y) ???????????????? ????
? z f1(x,y)
59
Procedure for Finding Limits of Integration in
Triple Integrals
3. ?? ??? ????????? D ??????????????????????? ??
?????? (???????????????? XY) ?????????????????????
????????????????????????????(????????? ??? y)
???????????? ????????????????????????????
(???????????? x) ???????????
????????????????????????? y g1(x)
??????????????????????? y g2(x)
4. ???????????????????????????????????????? ??????
? (????????? ??? x) ???????????????????? ?????????
????????? ???????????? x ??? a
???????????? x ??? b
?????? D ???????????? XY (????????????? Top view
??? D)
???????????????????????
60
Example1 Finding Limits of Integration in Triple
Integrals
???????? ??????????????????????
??????????????? Region D ??????????????????????
(0,0,0) (0,1,0) (1,1,0) ??? (0,1,1)
???????
??????
?????? 1. ???????????????????????????????????
??????????????????? (0,0,0) (0,1,1) ??? (1,1,0)
1.1 ?? Normal vector
1.2 ?????????????
2. ?????????????????????? ???????? ?????????? dy
dz dx
3. ??????????? y ????? function ??? x ??? z
??? ??????? ?????????
??? ?????? ?????????
61
Example1 Finding Limits of Integration in Triple
Integrals (cont.)
4. ??????????? z ????? function ??? x
z
B(0,1,1)
???????? ??? ????????????? ABC ??????????????
(1,1,0),(0,1,1) ??? (0,0,0) ???????????? XZ
D
y
?????? ?????????????????????????????????? 3
?????????????? x z 1, z 0 ??? z 0
C
(0,1,0)
x
????????????????????? z ????
A(1,1,0)
?????
???????
5. ??????????? x ??? ???????????? x ??? x 0
??????????????? x ??? x 1
????????????????????????????
62
Example2 Volume of a Region in Space
??????????????? f(x,y,z) 1 ????? Triple
integral ??????????????? Region
????????? ???????? ???????????????????????????????
?? z x23y2 ??? z 8-x2-y2
1. ??????????? z
????????????? Paraboloid 2 ???????????
????????????????
??????????????????
????????
2. ???????? Region ????????? ????? XY
z
?????????????????????????????????? ???????
???????????????????????? (???????????????????????)
x
y
??????????
????????
63
Example2 Volume of a Region in Space (continued)
?????
y
3. ?????????????? Region ????????? ????? XY
?????????
x
2
-2
???????? y ????? function ??? x
???????
?????
???????
4. ??????????? x ???
???????????????????????
64
Example2 Volume of a Region in Space (continued)
5. ????????????
65
Example3 Finding Limits of Integration in Triple
Integrals
??????????????????????????????????????????????
?????? ??????????????????????????????? 6
????????????????????????
a)
b)
d)
c)
f)
e)
66
Example4 Volume of a Region in Space
?????????????? Region ?????????????? 2 Cylinder
??????
??????????????????????????????? (?????????? z)
??????????????????????? (?????????? y)
?????? 1. ???????????????????????? dz dy dx
2. ??????????? z ???????
3. ??????????? y ???????
4. ??????????? x ???????
?????????????????????????????
67
Average Value in Space
????????????????? f(x,y,z) ?? Domain D
????????????????
???????? ???????????????? F(x,y,) xyz
?????????????????????
?????? 1. ???????????????????? ???
D
2. ??????????????
3. ????????????????? Volume ?????x???x???
238
???????????? 8/8 1
68
Moments and Mass in 3 Dimensions
?????????????????????????????? Triple integrals
???????? Moment ?????????????????? ???????? 3
???????????????????
??? ??????????????????????
? 3 ???? (??????????? ?????????????????)
??????????????????????????
Center of Mass (???????????????)
First order moments
????????????? yz
????????????? xz
????????????? xy
????? 3 ????
69
Moments and Mass in 3 Dimensions (continued)
Moment of inertia (second order moment)
??????????? x
??????????? y
??????????? z
Moment of inertia about a line L
??? r ???????????????? (x,y,z) ?????????? L
Radii of Gyration (???????????????)
??????????????? L
???????????????? x
???????????????? y
???????????????? z
70
Example Moments and Mass in 3 Dimensions
????????? Ix Iy ??? Iz ???????????????????????????
???????? d
??????
????????? (y2z2) ????????????????????????????????
???????????????
??????????????? ??????
??? M ???????????
71
Example Moments and Mass in 3 Dimensions
??????????????????????????????????????????????????
???? d
??????
1. ?????? ????????????????????????????????
z ???????????????????
2. ?????
?????????????? ????????????? Rectangular ????
Polar form ?????
??????????????? ??????
?????
72
Coordinate Systems
?????? Cartesian (Rectangular) Coordinate
?????????? (x,y,z) ??????????????????????? Space
3 ???? ???????????????????????? ??????????????????
?????????????????????????????????????? ?????????
????????? ???????????????????????????????????????
????????????? ?????????? ?????? Rectangular
Coordinate ??????????????????????
??????????????????????????????????????????????????
???????????????????? Polar coordinate ?????????
????????? Space 2 ???? ???? Cylindrical
Coordinate ??? Spherical Coordinate ?????????
Space 3 ???? ??????? ?????????????????????????????
?????? ?????????????????????????????????? ????????
??????? Coordinate ??????????
73
Cylindrical Coordinate
???????????????????????????????????? Polar
coordinate ??? Rectangular Coordinate ??? (x,y)
?????? rectangular ???????????? (r,q) ??????
Polar ??????? z ???? 2 ???????????????????
??????? ???? Cylindrical coordinate
???????? ??????? 3 ?????? r,q, ??? z ??????
????????????????????? ??????????? ???????????
3 ???????????????????????????? ?????????
(?????????????????????? z)
????????????????????????????????
xy ????????????????? Polar form (r,q)
???? ??????????????? z ??????
74
Relation Between (r,q, z ) and (x,y,z)
?????????????????? Cylindrical ??? Rectangular
coordinate ??????????
???????? z ?? Cylindrical ??? Cartesian
???????????????
75
Objects of Constant r,q, z in Cylindrical
Coordinate
?????????????????????????????????????????????
????? ?????????????? ????????????????????????????
????? ???????????????????????? ???????????????????
1. ?????? r r0 ???????????????????? ????????????
????? z ????????????????????? ??????? r0
2. ?????? z z0 ???????????????????? ???????? xy
???????????????????? xy ?????????? z0
3. ?????? q q0 ???????????????? ???????????????
xy ??????????? ????? xz ??????? q0
76
dV in Cylindrical Coordinate
?????? Triple integrals ????????????????? dV
?????????????????? ????????? ?????????? ????????
??????????????????????????????????????????????????
??????? dV ?????????????
???????????????????????? ??????? dV
?????????????????? ????????????????????????????
?????????????????????????????????????????????????
77
Triple Integral in Cylindrical Coordinate
??????????????????????????????????
?????????????? ?????????? z ???? function ???
(r,q) ?????? r ???? function ??? q
?????????????????????????????
?????????????????????????????? dz dr dq
????????????????????????????????????????????
1. Finding Limit of z in Cylindrical Coordinate
?????? z ?????????????? ????????????????????
? z g2(r,q) ??????????????????????? z
g1(r,q)
78
Finding Limit of Integration in Cylindrical
Coordinate
2. Finding Limit of r in Cylindrical Coordinate
????????????????????????? z ????
????????????? 2 ?????? (r,q) ?????????????????
(r,q) ??????????????????????? ????????? (r,q) ??
Polar coordinate ????????
2.1 ???????? Region ?????????????? XY
3. Finding Limit of q
2.2 ??????????? r ????? function ??? q
???????????
?????????
??????????
??????????
???????
79
Example 1 Finding Limit of Integration in
Cylindrical Coordinate
?????? ???????????????????????????????????????????
?????? 1. ??????????? z ??????????????????????????
??
????????????????? z 2 - y ??????????????????? z
0
????????????? Cylindrical coordinate
???????????? (r,q,z) ?????????????? z
??????????????????????? (r,q)
?????
????????????????? ???????????????????
2.??????????????????????? r ??????????????????????
??? Top view ????????????
??? Top view
???????? ???????????? r 0 ??????????????????????
???????? (0,0) ???
???????????????????????? x 1 ???????????????????
?????????????? ????????????? r ??? q
??????????
????? x 1 ?????
???????????????????
80
Example 1 Finding Limit of Integration in
Cylindrical Coordinate (cont.)
3. ??????????? q
???????????
??????????
4. ?????????????????????????
81
Example2 Triple Integral in Cylindrical
Coordinate
??????????????????????????????????????????????????
????????????????
?????????????????
??????????????????? z 0 ??????????????????????
???????????????????????????????? (0,1,0)
??????
1. ??????????? z ??????????????
????????????????????? ?????????????????????
??
2. ???????????????????????? XY
?????????????????????????????????????????????????
???????????????????? ????????????????????????????
?????????? 1 ?????????????????????????
(0,1,0) ??????????????? ??????????????????
Polar form ??????
????
82
Example2 Triple Integral in Cylindrical
Coordinate (cont.)
3. ??????????? r ????? function ??? q
???????? ????????? ?????
??????? (0,0) ??????? ??????????????? r
0 ???????????????????
?
4. ??????????? q
??????????????????????? x ???? ????????????????
0 ????????????? p
??????????????? ??????? xy
????????????????????????????
83
Example3 Triple Integral in Cylindrical
Coordinate
??????????????????????????????????????????????????
????????
????????????? ????
?????????????????????? xy ???????????????? 1
?????? 1. ?????????????????????????????? z
??????????
???????????????????????
2. ??????????? z ????? function ??? r ??? q
????????????????????
??????????????????????
3. ??????????? r
???????????? r 0
????????????? r 2
4. ??????????? q ???
???????????????????????
84
Example3 Triple Integral in Cylindrical
Coordinate (cont.)
5. ?????????????????
6. ????? Mxy
???
??????
85
Spherical Coordinate
??????????????????????????????????????????????????
????????????????????????????? ???????????????? 3
?????? r, q ??? f ?????????????????
r ?????????? (?????) ???????????? (0,0,0) ??????
(x,y,z)
q ????????????? xy ??????????????
??????????? x ???
f ?????????????????
??????????? z ???
?????????????????????????????????????? ???????????
??????????????? Latitude ??? Longitude ???
Altitude ???????? Latitude ??????????? q
Longitude ??????????? f ???? Altitude
?????????????????????? ??????????????? (?????????
r ????????????????? ???????????????)
86
Relation between r, q, f and other coordinates
?????????????????? Spherical ??? Rectangular
coordinate ??????????
From Cartesian to Spherical
From Spherical to Cartesian
?????????????????????????????????????? ???????????
???????????? ????????
???????? q ???????????????????? ??????????????????
?????????????????
87
Objects of Constant r, q, f in Spherical
Coordinate
?????????????????????????????????????????????
????? ?????????????? ????????????????????????????
????? ???????????????????????? ???????????????????
1. ?????? f f0 ?????????????????? ????? z
??????????????????????????????????? f0
?????????????? z ??????
2. ?????? q q0 ?????????????????????????? ??????
???????????? xy ?????????? z ??? ????????????? xz
??????? q0 ??????
3. ?????? r r0 ?????????????????????? ??????????
????? (0,0,0) ????????????????? r0
88
Example 1 Objects in Spherical Coordinate
??????????????????
?????????????????
?????? ??????????????????????????????? (0,0,1)
????????????????????????? Spherical Coordinate
??????????????? x,y,z ?????????????
???
z
(0,0,1)
???
????
y
x
89
Example 2 Objects in Spherical Coordinate
????????????????????????????????????
?????? ??????????????? x,y,z ?????????????
???
z
???
????
??? f ???????? cos f ??? sin f ???????????????
y
x
??????????? ?????????????????
90
dV in Spherical Coordinate
?????? Triple integrals ????????????????? dV
?????????????????? ????????? ?????????? ????????
??????????????????????????????????????????????????
???? dV ?????????????
????????????????????? ??????? dV
?????????????????? ????????????????????????????
??????????????????????????????????????????????
91
Triple Integral in Spherical Coordinate
???????????????????????????????
?????????????? ?????????? r ???? function ???
(q,f) ??????????????? f ??? q ????????????
?????????????????????????????
?????????????????????????????? dr df dq
????????????????????????????????????????????
1. Finding Limit of r in Spherical Coordinate
?????? r ?????????????? ????????????????????
r g2(q,f) ??????????????????? r
g1(q,f)
92
Finding Limit of Integration in Spherical
Coordinate
2. Finding Limit of f in Spherical Coordinate
????????? f ??????????? ??????????????????????????
? f ?????
3. Finding Limit of q in Spherical Coordinate
????????? q ??????????? ????????? ???????????????
??? q ?????
????????????
( ?????????? ????
?????????????? ?????????????????????????????
!)
93
Example 3 Triple Integral in Spherical Coordinate
??????????????????????????????????????????????????
??? r 1 ?????????? f p/3 ????????
?????? ?????????????????????????? ??????????????
?????????????????????????????
????????????? ???????????????? r 1
??????????????? r 0
?????????????? f ????
?????????????? q ????
??????
94
Example 3 Triple Integral in Spherical
Coordinate (cont.)
???
????????????????? r ???
????????????????? f ???
????????????????? q ???
95
Example 4 Triple Integral in Spherical Coordinate
???? Moment of inertia ???????????????????????????
????????? z ???????????????????????? ?????????????
???? 1
?????? ???? Moment of inertia about z-axis ???
???????????????????????
?????????????????
???????????????????????????????
96
Example 4 Triple Integral in Spherical
Coordinate (cont.)
????????????????? r ???
????????????????? f ???
????????????????? q ???
97
Formula for 3-D Coordinate Systems
From Spherical to Cartesian
dV
From Spherical to Cylindrical
98
Substitutions in Multiple Integrals
?????????????????????????????????????????????
?? (Coordinate system) ???? ??? (x,y,z) ??
Rectangular coordinate?????? (r,q,z) ??
Cylindircal coordinate ??????????????????????????
?????????????????? ????????????????????????
??????????? Rectangular, Cylindrical ???
Spherical Coordinate systems ????????????????????
?????????????? ????????????? ?????????????????
??????????? ?????????????????????????????????????
????????????????????????????? ?????
Substitution ?????? ????????? ????????????????????
?????????????????? ????????? ?????????????????????
?????????????????????????????????????????????????
? ????????????????????
99
Substitutions in Double Integrals
????????????????????????????????
????????????????????? (x,y) ????????????? (u,v)
??????????????????????
?????????????????????????????? 1 ??? 1 ???????? 1
????? uv-plane ?????????? 1 ????? xy-plane (??? f
??? g ??????? 0 ??????????????)
????????????????? x ??? y ??? f(x,y)
?????????????????????????????? u ???
v??????????????? f(g(u,v),h(u,v))
100
Example 1 Substitutions in Double Integrals
??????????????? ???????? Region G ?? uv-plane
???????????????? Region R ?? xy-plan
????????
101
Example 1 Substitutions in Double Integrals
????????????? ??? (u,v) ???? (x,y)
?????????????????????????????
????????????? ??? (x,y) ???? (u,v)
102
Substitutions in Double Integrals
????????????????????????????? f(x,y) ??
Region R ?? xy-plan ????????????????????
f(g(u,v),h(u,v) ?? Region G ?? uv-plane
?????????
??? ??? Jacobian
determinant ??????????????? Jacobian ??????????
????????????
103
Double Integral in Polar Form
???
??????????????????????? rq-plane ????
104
Example 2 Double Integral in Polar Form
????????
?????????????????????????????????? xy-plane
?????????????????
???????????????????????????? Polar coordinate
y
2
1
????????????????????????????????????? rq-plane
????
x
0
??????????????????????? rq-plane ????????????? ??
xy-plane ??? ????????????????????????? rq-plane
???????????????????????
105
Example 3 Substitutions in Double Integrals
???????? ?????????????
?????? ??????????? ????????????????????????? xy
?????????? ???????????????????????????????????????
??? ???????????
1. ???????????????? ????????????????
2. ??????????????????????????? uv ????????????
????????????? x ????
???????
??????????? x uv ??? y 2v ??????
????
106
Example 3 Substitutions in Double Integrals
(cont.)
2. ??????????????????????????? uv ????????????
(???)
????????????? x ????
??????
??????????? x uv ??? y 2v ??????
????
????????????? y ????
???????
??????
?????
????????????????????????????????? uv ??????
3. ????? Jacobian determinant
107
Example 3 Substitutions in Double Integrals
(cont.)
4. ????????? ???????????? f(x,y) ??????
f(g(u,v),h(u,v)) ???????????? ?????
5. ?????????
???
6. ?????
???
108
Example 4 Substitutions in Double Integrals
?????????????
?????? 1. ????????????????????????
??????? xy ????????????
2. ???????????????? ????????????????
3. ??????????????????????????? uv ????????????
3.1 ??????????????????????
??????????? ????????
????
3.2 ??????????????????????
??????????? ????????
????
109
Example 4 Substitutions in Double Integrals
(cont.)
3. ??????????????????????????? uv ????????????
(???)
3.3 ????????????????????
??????????? ????????
????
????????????????????????????????? uv ??????????
4. ????? Jacobian determinant
110
Example 4 Substitutions in Double Integrals
(cont.)
5. ????????? ???????????? f(x,y) ??????
f(g(u,v),h(u,v)) ???????????? ?????
6. ?????????
???
7. ?????
???
111
Substitutions in Triple Integrals
?? section ???????????????????????????????????????
??????????????????????? ?????????? ???????????????
??????????????????? 3 ???? ???????????????????????
??????????? 3 ?????????? ????????
??????????????????????????
?????
?????????????????????????
??? D ??????????????????????????? xyz ??? G
??????????????????????????? uvw
112
Jacobian Determinant in 3D
???? Jacobian determinant ?????? 3
????????????????
???????? ??????????????????????????????? xyz
??????????????????? rqz ??????????
?????
113
Example 5 Substitutions in Triple Integrals
??????????????????????????????????????????????????
??????????????????????????
114
Example 6 Substitutions in Triple Integrals
??????????????????????????????????????????????????
?????????
?????
115
Example 6 Substitutions in Triple Integrals
(cont.)
??????????????????????????????????????????????????
????????????????????????
116
Example 7 Substitutions in Triple Integrals
??????????
?????? 1.??????????????????????????????
xyz ??????????????
2. ??????????????????????????
?????????????
117
Example 7 Substitutions in Triple Integrals
(cont.)
3. ?????????????????????????? uvw
?????????????????
3.1 ?????
??????
????
3.2 ?????
??????
????
3.3 ????? y 0 ?????? v 0
3.4 ????? y 4 ?????? v 2
3.5 ????? z 0 ?????? w 0
3.6 ????? z 3 ?????? w 1
118
Example 7 Substitutions in Triple Integrals
(cont.)
???????????????????????????????? uvw ??????????
4. ????? Jacobian determinant
5. ????????? ???????????? F(x,y,z) ??????
F(g(u,v,w),h(u,v,w),k(u,v,w))
6. ?????????
119
Example 7 Substitutions in Triple Integrals
(cont.)
7. ?????
???
120
Exercise
1. ??????????????? u x 2y ??? v x y
???????????????????
2. ??? R ????????????? Quadrant ??? 1 ??????? xy
????????????????????????? xy 1 ??? xy 9
?????????? y x ??? y 4x ????????????????? x
u/v ??? y uv ????? u gt 0 ??? v gt 0
????????????????????
3. ???????????????????
?????????????????
???????? x au, y bv, ??? z cw