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ESSENTIAL CALCULUSESSENTIAL CALCULUS
CH12 Multiple CH12 Multiple integralsintegrals
In this Chapter: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 Coordinates
12.8 Change of Variables in Multiple Integrals
Review
Chapter 12, 12.1, P665
(See Figure 2.) Our goal is to find the volume of S.
}),(),,(0),,{( 3 RyxyxfzRzyxs
Chapter 12, 12.1, P667
Chapter 12, 12.1, P667
Chapter 12, 12.1, P667
5. DEFINITION The double integral of f over the rectangle R is
if this limit exists.
ij
m
i
n
jRyx
AjyijxifdAyxfii
),(lim),(
1
*
1
*
0,max
Chapter 12, 12.1, P668
AyxfdAyxfm
i
n
jji
Rnm
),(lim),(
1 1,
Chapter 12, 12.1, P668
If f (x, y) ≥ 0, then the volume V of the solid that lies above the rectangle R and below the surface z=f (x, y) is
R
dAyxfv ),(
Chapter 12, 12.1, P669
MIDPOINT RULE FOR DOUBLE INTEGRALS
where xi is the midpoint of [xi-1, xi] and yj is the midpoint of [yj-1, yj ].
AyxfdAyxf ji
m
i
n
jR
),(),(1 1
Chapter 12, 12.1, P672
10. FUBINI’S THEOREM If f is continuous on the rectangle R={(x, y} │a ≤ x≤ b, c ≤ y ≤d} , then
More generally, this is true if we assume that f is bounded on R, is discontinuous only on a finite number of smooth curves, and the iterated integrals exist.
b
a
d
c
d
c
b
aR
dxdyyxfdydxyxfdAyxf ),(),(),(
Chapter 12, 12.1, P673
R
b
a
d
cdyyhdxxgdAyhxg )()()()( where R ],[],[ dcbaR
Chapter 12, 12.2, P676
Chapter 12, 12.2, P676
Chapter 12, 12.2, P677
Chapter 12, 12.2, P677
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
),(),(
Chapter 12, 12.2, P678
Chapter 12, 12.2, P678
D
d
c
yh
yhdxdyyxfdAyxf
)(
)(
2
1
),(),(
where D is a type II region given by Equation 4.
Chapter 12, 12.2, P681
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, then
8.
If D=D1 U D2, where D1 and D2 don’t overlap except perhaps on their boundaries (see Figure 17), then
D D D
dAyxfdAyxfdAyxf1 2
),(),(),(
D
DAdA )(1
Chapter 12, 12.2, P682
Chapter 12, 12.2, P682
11. If m≤ f (x, y) ≤ M for all (x ,y) in D, then
D
DMAdAyxfDmA )(),()(
Chapter 12, 12.3, P684
Chapter 12, 12.3, P684
Chapter 12, 12.3, P684
Chapter 12, 12.3, P684
r2=x2+y2 x=r cosθ y=r sinθ
Chapter 12, 12.3, P685
Chapter 12, 12.3, P686
2. CHANGE TO POLAR COORDINATES IN A DOUBLE INTEGRAL If f is continuous on a polar rectangle R given by 0≤a≤r≤b, a≤θ≤β, where 0≤β-α≤2 , then
R
b
ardrdrrfdAyxf
)sin,cos(),(
Chapter 12, 12.3, P687
3. If f is continuous on a polar region of the form
then
)()(,),{( 21 hrhrD
D
h
hrdrdrrfdAyxf
)(
)(
2
1
)sin,cos(),(
Chapter 12, 12.5, P696
3. DEFINITION The triple integral of f over the box B is
if this limit exists.
B
l
i
m
jijkijkijk
n
kijk
zyxVzyxfdVzyxf
kii 1 1
**
1
*
0,,max),,(lim),,(
Chapter 12, 12.5, P696
B
l
i
m
jkj
n
ki
nmlVzyxfdVzyxf
1 1 1,,
),,(lim),,(
Chapter 12, 12.5, P696
4. FUBINI’S THEOREM FOR TRIPLE INTEGRALS If f is continuous on therectangular box B=[a, b]╳ [c, d ]╳[r, s], then
B
s
r
d
c
b
adxdydzzyxfdVzyxf ),,(),,(
Chapter 12, 12.5, P697
dAdzzyxfdVzyxfE D
yxu
yxu
),(
),(
2
1
),,(),,(
Chapter 12, 12.5, P697
b
a
xg
xg
yxu
yxuE
dzdydxzyxfdVzyxf)(
)(
),(
),(
2
1
2
1
),,(),,(
Chapter 12, 12.5, P698
d
c
xh
xh
yxu
yxuE
dzdxdyzyxfdVzyxf)(
)(
),(
),(
2
1
2
1
),,(),,(
Chapter 12, 12.5, P698
dAdxzyxfdVzyxfE D
yxu
yxu
),(
),(
2
1
),,(),,(
Chapter 12, 12.5, P699
dAdyzyxfdVzyxfE D
yxu
yxu
),(
),(
2
1
),,(),,(
Chapter 12, 12.5, P700
E
dVEV )(
Chapter 12, 12.6, P705
Chapter 12, 12.6, P705
To convert from cylindrical to rectangular coordinates, we use the equations
1 x=r cosθ y=r sinθ z=z
whereas to convert from rectangular to cylindrical coordinates, we use
2. r2=x2+y2 tan θ= z=zx
y
Chapter 12, 12.6, P706
Chapter 12, 12.6, P706
Chapter 12, 12.6, P706
Chapter 12, 12.6, P707
formula for triple integration in cylindrical coordinates.
E
h
h
rru
rrurdzdrdzrrfdVzyxf
)(
)(
)sin,cos(
)sin,cos(
2
1
2
1
),sin,cos(),,(
Chapter 12, 12.7, P709
0p 0
Chapter 12, 12.7, P709
Chapter 12, 12.7, P709
Chapter 12, 12.7, P709
Chapter 12, 12.7, P710
Chapter 12, 12.7, P710
cossinpx sinsinpy cospz
Chapter 12, 12.7, P710
2222 zyxp
Chapter 12, 12.7, P711
FIGURE 8Volume element in sphericalcoordinates: dV=p2sinødpdΘd ø
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 spherical coordinates
Chapter 12, 12.8, P716
Chapter 12, 12.8, P717
Chapter 12, 12.8, P718
7. DEFINITION The Jacobian of the transformation T given by x= g (u, v) and y= h (u, v) is
u
y
v
x
v
y
u
x
v
y
u
yv
x
u
x
vu
yx
),(
),(
Chapter 12, 12.8, P719
9. CHANGE OF VARIABLES IN A DOUBLE INTEGRAL Suppose that T is a C1
transformation whose Jacobian is nonzero and that maps a region S in the uv-plane onto a region R in the xy-plane. Suppose that f is continuous on R and that R and S are type I or type II plane regions. Suppose also that T is one-to-one, except perhaps on the boundary of . S. Then
R S
dudvvu
yxvuyvuxfdAyxf
),(
),()),(),,((),(
Chapter 12, 12.8, P721
Let T be a transformation that maps a region S in uvw-space onto a region R in xyz-space by means of the equations
x=g (u, v, w) y=h (u, v, w) z=k (u, v, w)
The Jacobian of T is the following 3X3 determinant:
12.
w
z
v
z
u
zw
y
v
y
u
yw
x
v
x
u
x
wvu
zyx
),,(
),,(
Chapter 12, 12.8, P722
R
dVzyxf ),,(13.
s
dudvdwwvu
zyxwvuzwvuywvuxf
),,(
),,()),,(),,,(),,,((
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