8/10/2019 Ch12 Multiple Integrals
1/59
ESSENTIAL
CALCULUS
CH12 Multipleintegrals
8/10/2019 Ch12 Multiple Integrals
2/59
In this Chapter:
12.1 Double Integrals over Rectangles
12.2 Double Integrals over General Regions
12.3 Double Integrals in Polar Coordinates
12.4 Applications of Double Integrals
12.5 Triple Integrals
12.6 Triple Integrals in Cylindrical Coordinates
12.7 Triple Integrals in Spherical Coordinates12.8 Change of Variables in Multiple Integrals
Review
8/10/2019 Ch12 Multiple Integrals
3/59
Chapter 12, 12.1, P665
(See Figure 2.) Our goal is to find
the volume of S.
}),(),,(0),,{( 3 RyxyxfzRzyxs
8/10/2019 Ch12 Multiple Integrals
4/59
Chapter 12, 12.1, P667
8/10/2019 Ch12 Multiple Integrals
5/59
Chapter 12, 12.1, P667
8/10/2019 Ch12 Multiple Integrals
6/59
Chapter 12, 12.1, P667
5. DEFINITION The double integral of f overthe rectangle R is
if this limit exists.
ij
m
i
n
jRyx
AjyijxifdAyxfii
),(lim),(
1
*
1
*
0,m ax
8/10/2019 Ch12 Multiple Integrals
7/59
Chapter 12, 12.1, P668
AyxfdAyxfm
i
n
j
ji
R
nm
),(lim),(
1 1
,
8/10/2019 Ch12 Multiple Integrals
8/59
Chapter 12, 12.1, P668
If f (x, y) 0, then the volume V of the solid thatlies above the rectangle R and below the surface
z=f (x, y) is
R
dAyxfv ),(
8/10/2019 Ch12 Multiple Integrals
9/59
Chapter 12, 12.1, P669
MIDPOINT RULE FOR DOUBLE INTEGRALS
wherexiis the midpoint of [xi-1,xi] and yjis the
midpoint of [yj-1, yj].
AyxfdAyxf ji
m
i
n
jR
),(),(1 1
8/10/2019 Ch12 Multiple Integrals
10/59
Chapter 12, 12.1, P672
10. FUBINIS THEOREM If f is continuous onthe rectangle R={(x, y} a x b, c y d} ,
then
More generally, this is true if we assume that fis bounded on R, is discontinuous only on afinite number of smooth curves, and theiterated integrals exist.
b
a
d
c
d
c
b
aR
dxdyyxfdydxyxfdAyxf ),(),(),(
8/10/2019 Ch12 Multiple Integrals
11/59
Chapter 12, 12.1, P673
R
b
a
d
c dyyhdxxgdAyhxg )()()()( where R ],[],[ dcbaR
8/10/2019 Ch12 Multiple Integrals
12/59
Chapter 12, 12.2, P676
8/10/2019 Ch12 Multiple Integrals
13/59
Chapter 12, 12.2, P676
8/10/2019 Ch12 Multiple Integrals
14/59
Chapter 12, 12.2, P677
8/10/2019 Ch12 Multiple Integrals
15/59
Chapter 12, 12.2, P677
8/10/2019 Ch12 Multiple Integrals
16/59
Chapter 12, 12.2, P677
3.If f is continuous on a type I region D such that
then
)}()(,),{( 21 xgyxgbxayxD
D
b
a
xg
xgdydxyxfdAyxf
2
1
),(),(
8/10/2019 Ch12 Multiple Integrals
17/59
Chapter 12, 12.2, P678
8/10/2019 Ch12 Multiple Integrals
18/59
Chapter 12, 12.2, P678
D
d
c
yh
yhdxdyyxfdAyxf
)(
)(
2
1
),(),(
where D is a type II region given by Equation 4.
8/10/2019 Ch12 Multiple Integrals
19/59
Chapter 12, 12.2, P681
8/10/2019 Ch12 Multiple Integrals
20/59
Chapter 12, 12.2, P681
D D D
dAyxgdAyxfdAyxgyxf ),(),()],(),([
D D
dAyxfcdAyxcf ),(),(
D g
dAyxgdAyxf ),(),(
6.
7.
If f (x, y) g (x, y) for all (x, y) in D, then8.
If D=D1U D2, where D1 and D2dont overlap
except perhaps on their boundaries (see Figure17), then
D D D
dAyxfdAyxfdAyxf
1 2
),(),(),(
8/10/2019 Ch12 Multiple Integrals
21/59
D
DAdA )(1
Chapter 12, 12.2, P682
8/10/2019 Ch12 Multiple Integrals
22/59
Chapter 12, 12.2, P682
11. If m f (x, y) M for all (x ,y) in D, then
D
DMAdAyxfDmA )(),()(
8/10/2019 Ch12 Multiple Integrals
23/59
Chapter 12, 12.3, P684
8/10/2019 Ch12 Multiple Integrals
24/59
8/10/2019 Ch12 Multiple Integrals
25/59
Chapter 12, 12.3, P684
8/10/2019 Ch12 Multiple Integrals
26/59
Chapter 12, 12.3, P684
r2=x2+y2 x=r cos y=r sin
8/10/2019 Ch12 Multiple Integrals
27/59
Chapter 12, 12.3, P685
8/10/2019 Ch12 Multiple Integrals
28/59
Chapter 12, 12.3, P686
2. CHANGE TO POLAR COORDINATES IN ADOUBLE INTEGRAL If f is continuous on a
polar rectangle R given by 0arb, a,where 0-2 , then
R
b
a
rdrdrrfdAyxf
)sin,cos(),(
8/10/2019 Ch12 Multiple Integrals
29/59
Chapter 12, 12.3, P687
3. If f is continuous on a polar region of the form
then
)()(,),{( 21 hrhrD
Dh
h rdrdrrfdAyxf
)(
)(
2
1 )sin,cos(),(
8/10/2019 Ch12 Multiple Integrals
30/59
Chapter 12, 12.5, P696
3. DEFINITION The triple integral of f overthe box B is
if this limit exists.
B
l
i
m
j
ijkijkijk
n
k
ijkzyx
VzyxfdVzyxfkii 1 1
**
1
*
0,,m ax),,(lim),,(
8/10/2019 Ch12 Multiple Integrals
31/59
Chapter 12, 12.5, P696
B
l
i
m
j
kj
n
k
inml
VzyxfdVzyxf1 1 1
,,),,(lim),,(
8/10/2019 Ch12 Multiple Integrals
32/59
Chapter 12, 12.5, P696
4. FUBINIS THEOREM FOR TRIPLEINTEGRALS If f is continuous on therectangular box B=[a, b][c, d ][r, s], then
B
s
r
d
c
b
adxdydzzyxfdVzyxf ),,(),,(
8/10/2019 Ch12 Multiple Integrals
33/59
Chapter 12, 12.5, P697
dAdzzyxfdVzyxfE D
yxu
yxu
),(
),(
2
1
),,(),,(
8/10/2019 Ch12 Multiple Integrals
34/59
Chapter 12, 12.5, P697
b
a
xg
xg
yxu
yxuE
dzdydxzyxfdVzyxf)(
)(
),(
),(
2
1
2
1
),,(),,(
8/10/2019 Ch12 Multiple Integrals
35/59
Chapter 12, 12.5, P698
d
c
xh
xh
yxu
yxuE
dzdxdyzyxfdVzyxf)(
)(
),(
),(
2
1
2
1
),,(),,(
8/10/2019 Ch12 Multiple Integrals
36/59
Chapter 12, 12.5, P698
dAdxzyxfdVzyxfE D
yxu
yxu
),(
),(
2
1
),,(),,(
8/10/2019 Ch12 Multiple Integrals
37/59
Chapter 12, 12.5, P699
dAdyzyxfdVzyxfE D
yxu
yxu
),(
),(
2
1
),,(),,(
8/10/2019 Ch12 Multiple Integrals
38/59
Chapter 12, 12.5, P700
E dVEV )(
8/10/2019 Ch12 Multiple Integrals
39/59
Chapter 12, 12.6, P705
8/10/2019 Ch12 Multiple Integrals
40/59
Chapter 12, 12.6, P705
To convert from cylindrical to rectangularcoordinates, we use the equations
1 x=r cos y=r sin z=z
whereas to convert from rectangular to cylindrical
coordinates, we use
2. r2=x2+y2 tan = z=zx
y
8/10/2019 Ch12 Multiple Integrals
41/59
Chapter 12, 12.6, P706
8/10/2019 Ch12 Multiple Integrals
42/59
Chapter 12, 12.6, P706
8/10/2019 Ch12 Multiple Integrals
43/59
Chapter 12, 12.6, P706
8/10/2019 Ch12 Multiple Integrals
44/59
8/10/2019 Ch12 Multiple Integrals
45/59
Chapter 12, 12.7, P709
0p 0
8/10/2019 Ch12 Multiple Integrals
46/59
Chapter 12, 12.7, P709
8/10/2019 Ch12 Multiple Integrals
47/59
Chapter 12, 12.7, P709
8/10/2019 Ch12 Multiple Integrals
48/59
Chapter 12, 12.7, P709
8/10/2019 Ch12 Multiple Integrals
49/59
Chapter 12, 12.7, P710
8/10/2019 Ch12 Multiple Integrals
50/59
Chapter 12, 12.7, P710
cossinpx sinsinpy cospz
8/10/2019 Ch12 Multiple Integrals
51/59
8/10/2019 Ch12 Multiple Integrals
52/59
8/10/2019 Ch12 Multiple Integrals
53/59
Chapter 12, 12.7, P711
where E is a spherical wedge given by
E
dVzyxf ),,(
d
c
b
addpdppsomppf
sin)cos,sin,cossin( 2
},,),,{( dcbpapE
Formula for triple integration in sphericalcoordinates
8/10/2019 Ch12 Multiple Integrals
54/59
Chapter 12, 12.8, P716
8/10/2019 Ch12 Multiple Integrals
55/59
Chapter 12, 12.8, P717
8/10/2019 Ch12 Multiple Integrals
56/59
9 CHANGE OF VARIABLES IN A DOUBLE
8/10/2019 Ch12 Multiple Integrals
57/59
Chapter 12, 12.8, P719
9. CHANGE OF VARIABLES IN A DOUBLEINTEGRAL Suppose that T is a C1
transformation whose Jacobian is nonzero and
that maps a region S in the uv-plane onto aregion R in the xy-plane. Suppose that f iscontinuous on R and that R and S are type I ortype II plane regions. Suppose also that T is
one-to-one, except perhaps on the boundary of .S. Then
R S dudvvu
yx
vuyvuxfdAyxf ),(
),(
)),(),,((),(
8/10/2019 Ch12 Multiple Integrals
58/59
8/10/2019 Ch12 Multiple Integrals
59/59
Top Related