chapter 5 multiple integrals; applications of integration ( 다중적분 ; 적분의 응용 )...
TRANSCRIPT
Chapter 5 Multiple integrals; applications of integration ( 다중적분 ; 적분의 응용 )
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Lecture 16 Double & Triple integrals
- Use for integration : finding areas, volume, mass, moment of inertia, and so
on.
- Computers and integral tables are very useful in evaluating integrals.
1) To use these tools efficiently, we need to understand the notation and
meaning of integrals.
2) A computer gives you an answer for a definite integral.
1. Introduction
b
a
b
adxxfydx )(
AREA under the curve
AA
dxdyyxfdAyxf ),(),(
VOLUME under the surface “double integral”
2. Double and triple integrals ( 이중 , 삼중 적분 )
Example 1.
- Iterated integrals
AAA
dxdyydxdyzdAzV )1()(
AAA
dydxydxdyydxdyzV )1()1()(
2
22
0
222
0
22
0
264)2
()1( xxy
ydyyzdyxx
y
x
y
1
0
21
0
22
0 3
5)264(
xx
x
yA
dxxxdxzdyzdydx
2
0
2
0
2/1
0
2
0
2/1
0
3
5)2/1)(1(
)1()1(
y
y
y
y
y
xA
dyyy
dyyxdydxyzdxdy
(a)
(b)
‘Integration sequence does not matter.’
12 yx
Integrate with respect to y first,
A
b
ax
xy
xyy
dxdyyxfdxdyyxf)(
)(
2
1
),(),(
Integrate with respect to x first,
A
d
cy
yx
yxx
dydxyxfdxdyyxf)(
)(
2
1
),(),(
Integrate in either order,
dydxyxfdxdyyxfdxdyyxfd
cy
yx
yxxA
b
ax
xy
xyy
)(
)(
)(
)(
2
1
2
1
),(),(),(
In case of ),()(),( yhxgyxf
A
b
ax
d
c
b
a
d
cy
dyyhdxxgdydxyhxgdxdyyxf )()()()(),(
Example 2. mass=?
density
f(x,y)=xy
(0,0)
(2,1)
xydxdydxdyyxfdM ),(
11
0
2
0
2
0
1
0
yx xyA
ydyxdxxydxdydMM
Triple integral f(x,y,z) over a volume V, VV
dxdydzzyxfdVzyxf ),,(),,(
Example 3. Find V in ex. 1 by using a triple integral,
1
0
22
0
1
0
22
0
1
0
)1(x
x
yV x
x
y
y
z
dydxydydxdzdxdydz
Example 4. Find mass in ex. 1 if density =x+z,
dxdydzzxdM )(
2}1)23{(6/1}1)23{(2
2/)1()1(
)2
(
)(
1
0
32
1
0
22
0
2
1
0
22
0
1
0
2
1
0
22
0
1
0
dxxxx
dydxyyx
dydxz
xz
dydxdzzxdMM
x
x
x
y
x
x
y
y
z
x
x
y
y
zV
3. Application of integration; single and multiple integrals ( 적분의 응용 ; 단일적분 , 다중적분 )
Example 1. y=x^2 from x=0 to x=1
(a) area under the curve
(b) mass, if density is xy
(c) arc length
(d) centroid of the area
(e) centroid of the arc
(f) moments of the inertia
(a) area under the curve3
1
3
1
0
31
0
21
0
xdxxydxA
xx
(b) mass, if density of xy
1
0
5
0
1
0
1
00 12
1
2
22
x
x
yx x
x
yA
dxx
ydyxdxxydxdydMM
2xy
0 1
(c) arc length of the curve
dydydxdxdxdydydxds
dydxds
2222
222
)/(1)/(1
ds
dx
dy
(d) centroid of the area (or arc)
dxxdsxdx
dy 241 ,2
4
)52ln(5241
1
0
2 dxxdss
cf. centroid : constant
dA
xdAxxdAdAx ,
, , zdAdAzydAdAyxdAdAx
2xy
10
3
10
1
10or ,
4
3
4
1
4or ,
1
0
51
0 0
1
0 0
1
0
41
0 0
1
0 0
22
22
yx
AyydydxdydxydAy
xx
AxxdydxdydxxdAx
x
x
yx
x
y
x
x
yx
x
y
In our example,
mass of centroid: xdMdMx
arc of centroid : dsxdsx
(e)
If is constant,
1
0
221
0
21
0
2
1
0
21
0
2
414141
4141
dxxxdxxydxxydsy
dxxxdxxxdsx
(f) moments of the inertia
dxdydzrdMdMlI )(for ,2
dxdydzyxdMyxI
dxdydzxzdMxzI
dxdydzzydMzyI
z
y
x
)()(
)()(
)()(
2222
2222
2222
80
7)(
,16
1
2)(
,40
1
4)(
1
0 0
22
1
0
1
0
7
0
21
0 0
22
1
0
1
0
9
0
21
0 0
22
2
22
22
x
yx
x
y
z
x
x
yx
x
y
y
x
x
yx
x
y
x
IIxydydxyxI
dxx
xydydxxxydydxxzI
dxx
xydydxyxydydxzyI
In our example, (=xy)
yxz IIIcf .
EX. 2 Rotate the area of Ex. 1 (y=x^2) about x-axis
(a) volume
(b) moment of inertia about x axis
(c) area of curved surface
(d) centroid of the curved volume
(a) volume
5
1
0
41
0
2 dxxdxyV(i)
(ii)
22
2424 to
xzx
zxyzxy
dxdydzV
1
0
2
2
24
24x
x
xz
zxy
zxy
dydzdxV
(b) I_x (=const.)
MdydzdxzydVzyIx
xz
xz
zxy
zxy
x 18
5
18)()(
1
0
2222
2
2
24
24
(c) area of curved surface
ydsdA 2
1
0
221
0
4122xx
dxxxydsA
(d) centroid of surface
1
0
2x
ydsxxdAAx
Chapter 5 Multiple integrals: applications of integration
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Lecture 17 Change of variables in integrals
4. Change of variables in integrals: Jacobians ( 적분의 변수변환 ; Jacobian)
In many applied problems, it is more convenient to use other coordinate
systems instead of the rectangular coordinates we have been using.
sin
cos
ry
rx
- polar coordinate:
dxdydA1) Area
rdrdrddr
2) Curve 222 dydxds 22 )( rddr
drdr
drdr
d
drds 2222 )(1)(
Example 1 r=a, density
(a) centroid of the semicircular area 0 . ycf
Ad
Axdx
2
00
2/
2/ 2ardrrdrd
dxdydA
a
r
a
r
3
22cos))(cos(
3
0
2
0
2/
2/
2
0
2/
2/
adrrdrdrrdrdrxdA
a
r
a
r
a
r
3
4
2
2
2
32 ax
aaxxdAdAx
(b) moment of inertia about the y-axis
8cos
)(
4
0
222/
2/
222222
ardrdr
rdrdxdxdyxdxdydzxdMxdMzxI
a
r
y
,2
2
0
2/
2/
ardrdrdrdM
a
r
48
2 24
2
Maa
a
MI y
- Cylindrical coordinate
- Spherical coordinate
22222
sin
cos
dzdrdrds
dzrdrddV
zz
ry
rx
2222222
2
sin
sin
cos
sinsin
cossin
drdrdrds
ddrdrdV
rz
ry
rx
Jacobians (Using the partial differentiation)
t
y
s
yt
x
s
x
ts
yx
ts
yxJJ
),(
),(
,
,
dsdtJdAdxdy
rr
ry
r
y
x
r
x
r
yx
cossin
sincos
),(
),( rdrddxdy
t
w
s
w
r
wt
v
s
v
r
vt
u
s
u
r
u
tsr
wvuJ
),,(
),,( drdsdtJtsrfdudvdwwvuf ),,(),,(
** Prove that ddrdrdV sin2
y
x
z
r=h
h
Example 2. ? and ? zIz
322
3
0
2
0 0
2
0
hdz
zdzrdrddVM
hh
z
z
r
4
3
43
,42
2
43
4
0
2
0 0
2
0
hz
hhz
hdz
zz
dzzrdrdzdVdVz
h
h
z
z
r
25
0
4
0 0
22
0 10
3
1042 Mh
hdz
zdzrdrdrI
hh
z
z
r
z
Mass:
Centroid:
Moment of inertia:
Example 3. Moment of inertia of ‘solid sphere’ of radius a
3
44
3
sin
33
2
0 0
2
0
aa
ddrdrdVMa
r
15
82
3
4
5
sin)sin()(
55
2
0 0
222
0
22
aa
ddrdrrdMyxIa
r
2
5
2MaI z
2222222
2
sin
sin
cos
sinsin
cossin.
drdrdrds
ddrdrdV
rz
ry
rxcf
Example 4. I_z of the solid ellipsoid 12
2
2
2
2
2
c
z
b
y
a
x
1''' then ,',',' 222 zyxczzbyyaxx
',',' cdzdzbdydyadxdx
1) radius of sphere of volume(''' abcdzdydxabcM
abcabcM 3
41
3
4 3
In a similar way,
')''()( 222222 dVybxaabcdVyxI
22222222 '''' where ,''3
1'''''' zyxrdVrdVzdVydVx
5
4''4
)''''sin'('''
1
0
4
2
0 0
221
0
2
drr
dddrrrdVrr
5
4
3
1)('''' 222222 baabcdVybdVxaabcI
)(5
1 22 baMI
5. Surface integrals (?) ( 표면적분 )
dxdydAdAdxdy sec ,cos ‘projection of the surface to xy plane’
dxdydA sec kncos
surface tonormal),,( grad z
ky
jx
izyx
.),,( constzyx
gradgradn /)(
cos/
grad
z
grad
gradkkn
z
zyx
z
grad
kn
/
)()()(
/
1
cos
1sec
222
1 so
),,(),,( ),,(For
z
yxfzzyxyxfz
1)()(cos
1sec 22
y
f
x
f
Example 1. Upper surface of the sphere by the cylinder
0 ,1 22222 yyxzyx
.),,( constzyx
222),,( zyxzyx
22
222
1
11)2()2()2(
2
1
/sec
yxzzyx
zz
grad
1 to0 from
to0 from 2
y
yyx
1
022
0 12
2
y
yy
x yx
dxdy
/2 0 from
sin to0 from
r
2)cos1(2)1sin1(2
121
2
2/
0
2/
0
2
2/
0
22/
0
2/
02
sin
0
dd
drr
rdrd
x
H. W. (due 5/28)
Chapter 5
2-43 3-17, 18, 19, 20 4-4