integral susun
TRANSCRIPT
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Chapter 5 Multiple integrals; applications of integration
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
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- 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
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b
a
b
adxxfydx )(
AREA under the curve
AA dxdyyxfdAyxf ),(),(
VOLUME under the surface
double integral
2. Double and triple integrals
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Example 1.
- Iterated integrals
AAA
dxdyydxdyzdAzV )1()(
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AAA
dydxydxdyydxdyzV )1()1()(
2
22
0
222
0
22
0
264)2
()1( xxy
ydyyzdy
xx
y
x
y
1
0
2
1
0
22
035)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
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Integrate with respect toyfirst,
A
b
ax
xy
xyy
dxdyyxfdxdyyxf
)(
)(
2
1
),(),(
Integrate with respect toxfirst,
A
d
cy
yx
yxx
dydxyxfdxdyyxf
)(
)(
2
1
),(),(
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Integrate in either order,
dydxyxfdxdyyxfdxdyyxf
d
cy
yx
yxxA
b
ax
xy
xyy
)(
)(
)(
)(
2
1
2
1
),(),(),(
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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 ),(
1
1
0
2
0
2
0
1
0
yx xyA
ydyxdxxydxdydMM
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Triple integralf(x,y,z) over a volume V, VV
dxdydzzyxfdVzyxf ),,(),,(
Example 3. Find Vin 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
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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
dxxx
x
dydxyyx
dydx
z
xz
dydxdzzxdMM
x
x
x
y
x
x
y
y
z
x
x
y
y
zV
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3. Application of integration; single and multiple integrals
Example 1.y=x^2fromx=0 tox=1
(a) area under the curve(b) mass, if density isxy
(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
2
1
0
xdxxydxA
xx
(b) mass, if density ofxy
1
0
5
0
1
0
1
0012
1
2
22
x
x
yx x
x
yA
dxx
ydyxdxxydxdydMM
2xy
0 1
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(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
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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,
massofcentroid: xdMdMx
arcofcentroid: dsxdsx
(e)
If is constant,
1
0
22
1
0
2
1
0
2
1
0
2
1
0
2
414141
4141
dxxxdxxydxxydsy
dxxxdxxxdsx
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(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
2
1
0 0
22
1
0
1
0
9
0
2
1
0 0
22
2
22
22
x
yx
x
y
z
x
x
yx
x
yy
x
x
yx
x
y
x
IIxydydxyxI
dx
x
xydydxxxydydxxzI
dxx
xydydxyxydydxzyI
In our example, (=xy)
yxz IIIcf .
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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
4
1
0
2 dxxdxyV(i)
(ii)
22
2424 to
xzx
zxyzxy
dxdydzV
1
0
2
2
24
24x
x
xz
zxy
zxy
dydzdxV
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(b) I_x (=const.)
MdydzdxzydVzyIx
xz
xz
zxy
zxy
x18
5
18)()(
1
0
2222
2
2
24
24
(c) area of curved surface
ydsdA 2
1
0
22
1
0
4122xx
dxxxydsA
(d) centroid of surface
1
0
2x
ydsxxdAAx
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Chapter 5 Multiple integrals: applications of integration
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
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4. Change of variables in integrals: Jacobians ( ; 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:
dxdydA 1) Area
rdrdrddr
2) Curve 222 dydxds
22 )( rddr
drdr
drdr
d
drds 2222 )(1)(
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Example 1 r=a, density
(a) centroid of the semicircular area 0. ycf
AdAxdx
2
00
2/
2/2 ardrrdrd
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
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(b) moment of inertia about the y-axis
8cos
)(
4
0
22
2/
2/
222222
ardrdr
rdrdxdxdyxdxdydzxdMxdMzxI
a
r
y
,2
2
0
2/
2/
ardrdrdrdM
a
r
48
2 24
2
Maa
a
MIy
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- Cylindrical coordinate
- Spherical coordinate
22222
sin
cos
dzdrdrds
dzrdrddVzz
ry
rx
2222222
2
sin
sin
cos
sinsin
cossin
drdrdrds
ddrdrdV
rz
ry
rx
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Jacobians (Using the partial differentiation)
t
y
s
yt
x
s
x
ts
yx
ts
yxJJ
),(
),(
,
,
dsdtJdAdxdy
rr
ry
r
y
x
r
x
ryx
cossin
sincos),(),( rdrddxdy
t
w
s
w
r
wtv
sv
rv
t
u
s
u
r
u
tsrwvuJ
),,(),,( drdsdtJtsrfdudvdwwvuf ),,(),,(
** Prove that ddrdrdV sin2
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y
x
z
r=h
h
Example 2. ?and? zIz
322
3
0
2
0 0
2
0
hdzzdzrdrddVM
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
2
2
010
3
1042 Mh
hdz
zdzrdrdrI
hh
z
z
r
z
Mass:
Centroid:
Moment of inertia:
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Example 3. Moment of inertia of solid sphere of radius a
3
44
3
sin
33
2
0 0
2
0
aa
ddrdrdVM
a
r
15
82
3
4
5
sin)sin()(
55
2
0 0
222
0
22
aa
ddrdrrdMyxI
a
r
2
5
2MaIz
2222222
2
sin
sin
cos
sinsin
cossin.
drdrdrds
ddrdrdV
rz
ry
rxcf
222
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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)radiusofsphereofvolume(''' abcdzdydxabcM
abcabcM 3
41
3
4 3
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In a similar way,
')''()(222222 dVybxaabcdVyxI
22222222 ''''where,''31'''''' zyxrdVrdVzdVydVx
5
4''4
)''''sin'('''
1
0
4
2
0 0
22
1
0
2
drr
dddrrrdVrr
54
3
1)('''' 222222
baabcdVybdVxaabcI
)(5
1 22 baMI
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5. Surface integrals (?)
dxdydAdAdxdy sec,cos projection of the surface to xy plane
dxdydA sec kn
cos
surfacetonormal),,(gradz
k
y
j
x
izyx
.),,( constzyx
gradgradn /)(
cos
/
grad
z
grad
gradkkn
z
zyx
z
grad
kn
/
)()()(
/
1
cos
1sec
222
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1so
),,(),,(),,(For
z
yxfzzyxyxfz
1)()(cos
1sec 22
y
f
x
f
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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
yxz
zyx
zz
grad
1to0from
to0from 2
y
yyx
1
022
0 12
2
y
yy
x yx
dxdy
/20from
sinto0from
r
2)cos1(2)1sin1(2
121
2
2/
0
2/
0
2
2/
0
2
2/
0
2/
02
sin
0
dd
drr
rdrd
x