Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Chapter 5 Multiple integrals; applications of integration
(다중적분 ; 적분의 응용)
Lecture 16 Double & Triple integrals
1. Introduction
- 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.
2. Double and triple integrals (이중, 삼중 적분)
b
b
a
a
 ydx 
f ( x)dx
AREA under the curve

A
f ( x, y )dA   f ( x, y )dxdy
A
VOLUME under the surface
“double integral”
- Iterated integrals
Example 1.
V   z dA   ( z)dxdy  (1  y)dxdy
A
A
A
2x  y  1
V   ( z)dxdy  (1  y)dxdy   (1  y)dydx
A
A
A
‘Integration sequence does not matter.’
22 x
(a)
2 2 x
2 2 x
y2
y0zdy  y0(1  y)dy ( y  2 )
0
 4  6x  2x2
1
2  2 x 
5
2
zdydx

zdy
dx

(
4

6
x

2
x
)
dx



A
 y0  x0
3
x 0 


1

(b)

2
1 y / 2

1 y / 2
zdxdy

(
1

y
)
dx
dy

x
(
1

y
)
dy


A
y 0  x0

0

y 0

2
2


y 0
(1  y )(1  y / 2)dy 
5
3
 Integrate with respect to y first,

A
 y2 ( x )

f ( x, y )dxdy     f ( x, y )dy dx

xa 
 y  y1 ( x )
b
 Integrate with respect to x first,

A
 x2 ( y )

f ( x, y )dxdy     f ( x, y )dxdy

y c 
 x  x1 ( y )
d
 Integrate in either order,

A
d  x2 ( y )
 y2 ( x )


f ( x, y)dxdy     f ( x, y)dy dx     f ( x, y)dxdy


xa 
y c 
 y  y1 ( x )
 x  x1 ( y )
b
 In case of

A
f ( x, y)  g ( x)h( y),
b
 d

f ( x, y )dxdy    g ( x)h( y )dydx    g ( x)dx   h( y )dy 
x  a y c
a
 c

b
d
Example 2. mass=?
(2,1)
density
f(x,y)=xy
dM  f ( x, y)dxdy  xydxdy

 2
 1

M   dM    xydxdy   xdx  ydy  1


A
x 0 y 0
 x 0  y 0 
2
(0,0)
1
 Triple integral f(x,y,z) over a volume V,
 f ( x, y, z )dV   f ( x, y, z )dxdydz
V
Example 3. Find V in ex. 1 by using a triple integral,
1 2 2 x


y 0  z0dz dydx  x0 y0(1  y)dydx


1 2  2 x 1 y
 dxdydz 
V
x 0
V
Example 4. Find mass in ex. 1 if density =x+z,
dM  ( x  z)dxdydz



M   dM      ( x  z )dz dydx
V
x 0 y 0  z 0

1 2  2 x
2 1 y 
z
    ( xz  ) dydx

2 z 0 
x 0 y 0 

1 2  2 x 1 y
1 22 x

  x(1  y)  (1  y) / 2dydx
2
x 0 y 0
x

2
3
{(
3

2
x
)

1
}

1
/
6
{(
3

2
x
)

1
}
x0  2
dx  2

1

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
y  x2
0
1
1
1
1
x3
1
2
A   ydx   x dx 

3 0 3
x 0
x 0
(a) area under the curve
(b) mass, if density of xy
 1 x5

 x
1
M   dM    xydxdy   xdx  ydy  
dx 


12
A
x 0 y 0
 x 0  y 0  x 0 2
1
x2
1
2
(c) arc length of the curve
y  x2
ds2  dx2  dy2
ds  dx  dy  1  (dy / dx) dx  1  (dx / dy) dy
2
2
2
2
ds
dy
dx
dy
 2 x,
dx
ds  1  4 x 2 dx
1
s   ds   1  4 x 2 dx 
0
2 5  ln(2  5 )
4
(d) centroid of the area (or arc)
 xdA   xdA,
xdA

x
 dA
 xdA   xdA ,
 ydA   ydA ,
cf. centroid : constant
 zdA   zdA
In our example,
1
x2
1
1
x2
 xdA    xdydx    xdydx, or
x 0 y 0
1
x2
x 0 y 0
1
x4
1
3
xA 
 x
4 0 4
4
1
x2
x5
1
3
yA 
 y
10 0 10
10
 ydA    ydydx    ydydx, or
x 0 y 0
(e)
x 0 y 0
 x dM   xdM : centroid of mass
 x ds   xds : centroid of arc
If  is constant,
1
1
2
x
ds

x
1

4
x
dx

x
1

4
x
dx



2
0
0
1
1
1
2
2
y
ds

y
1

4
x
dx

y
1

4
x
dx

x
1

4
x
dx




2
0
2
0
0
(f) moments of the inertia
I   l 2 dM ,
for dM   (r )dxdydz
I x   ( y 2  z 2 )dM   ( y 2  z 2 ) dxdydz
I y   ( z 2  x 2 )dM   ( z 2  x 2 ) dxdydz
I z   ( x 2  y 2 )dM   ( x 2  y 2 ) dxdydz
In our example, (=xy)
1
x2
1
x2
1
x2
1
x2
1
9
x
1
I x    ( y 2  z 2 ) xydydx   y 2 xydydx  dx  ,
4
40
x 0 y 0
x 0 y 0
0
1
x7
1
I y    ( z  x ) xydydx   x xydydx  dx  ,
2
16
x 0 y 0
x 0 y 0
0
2
1
Iz 

x2
2
2
2
2
(
x

y
) xydydx I x  I y 

x 0 y 0
7
80
cf . I z  I x  I y
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
1
(i)
1
V   y dx   x 4 dx 
2
0
(ii)
0

5
V   dxdydz
y   x 4  z 2 to y   x 4  z 2
 x2  z  x2
1
V
x2
 
y  x 4  z 2

x 0 z   x 2 y   x 4  z 2
dydzdx
(b) I_x
(=const.)
I x  ( y  z ) dV  
2
4
2
1 z  x 2 y  x  z
2
 
x 0 z   x 2 y  
(c) area of curved surface
dA  2yds
1
A
 2yds 
x 0
1
2
2
2

x
1

4
x
dx

x 0
(d) centroid of surface
1
x A   xdA   x  2yds
x 0

5
 4 2 ( y  z )dydzdx 18   18 M
x z
2
2
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Chapter 5 Multiple integrals: applications of integration
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.
- polar coordinate:
1) Area
x  r cos
y  r sin 
dA  dxdy
 dr  rd  rdrd 
2) Curve
ds2  dx2  dy2
 dr2  (rd )2
ds  (
dr 2 2
d
)  r d  1  r 2 ( ) 2 dr
d
dr
Example 1 r=a, density 
(a) centroid of the semicircular area
cf . y  0
xd A

x
dA
 dA   dxdy
a

 /2
  
a
rdrd   rdr 
r 0   / 2
a
r 0
 /2

2
a2
a
 /2
a
2a 3
 xdA  r0  (/ r2 cos )(rdrd )  r0  r/ 2 cos drd  r02r dr  3
2
x  dA   xdA 
a2
2a 3
4a
x 
 x
2
2
3
2
(b) moment of inertia about the y-axis
I y   ( x 2  z 2 )dM   x 2 dM   x 2 dxdydz  x 2 dxdy   x 2 rdrd

a
 /2
  
r cos  rdrd  
2
2
r 0  / 2
M    rdrd 
a
 /2
  
rdrd  
r 0   / 2
2M a 4 Ma 2
Iy  2

a 8
4
a 2
2
,
a 4
8
- Cylindrical coordinate
x  r cos
y  r sin 
zz
dV  rdrddz
ds2  dr2  r 2 d 2  dz2
- Spherical coordinate
x  r sin  cos
y  r sin  sin 
z  r cos
dV  r 2 sin drdd
ds2  dr 2  r 2 d 2  r 2 sin 2 d 2
Jacobians (Using the partial differentiation)
x
 x, y  ( x, y ) s
J  J


 s, t  ( s, t ) y
s
x
t
y
t
dxdy  dA  J dsdt
x
( x, y) r

(r , ) y
r
x
 
y

u
r
 (u , v, w) v
J

 ( r , s, t )
r
w
r
** Prove that
cos
r sin 
u
s
v
s
w
s
 r sin 
r
cos
u
t
v
t
w
t
dxdy rdrd
 f (u, v, w)dudvdw   f (r , s, t ) J drdsdt
dV  r 2 sin drdd
Example 2.
z  ? and I z  ?
z
r=h
Mass:
2
z2
h3
M   dV      rdrddz    2  dz 
2
3
z 0 r 0  0
0
Centroid:
h
z
h
z  dV   zdV 
h
z
y
2
  
h
zrdrddz
x
z 0 r 0 0
z2
h 4
 2  z dz 
,
2
4
0
h
z
h3
3

h 4
4
 z
2
z4
h5 3
I z     r rdrddz   2  dz 
 Mh2
4
10 10
z 0 r 0  0
0
h
Moment of inertia:
3h
4
z
h
2
Example 3. Moment of inertia of ‘solid sphere’ of radius a
M   dV  
2

a



 
r 2 sin drdd
0 0 r 0
a3
4  a 3
    4 
3
3
I   ( x  y )dM  
2
2
2

a
  
(r 2 sin 2  )r 2 sin drdd
0 0 r 0
a5 4
8a 5 
     2 
5 3
15
cf . x  r sin  cos
2
 I z  Ma 2
5
y  r sin  sin 
z  r cos
dV  r 2 sin drdd
ds2  dr 2  r 2 d 2  r 2 sin 2 d 2
Example 4. I_z of the solid ellipsoid
x2 y 2 z 2
 2  2 1
2
a
b
c
x  ax' , y  by' , z  cz' , thenx'2  y'2  z'2  1
dx  adx' , dy  bdy' , dz  cdz'
M  abc  dx ' dy ' dz '  abc  ( volume of sphere of radius 1)
4
4
M  abc   13  abc
3
3
In a similar way,
I   ( x 2  y 2 ) dV  abc  (a 2 x'2 b 2 y '2 )dV '
2
2
2
x
'
dV
'

y
'
dV
'

z
'


 dV ' 
 r ' dV ' 
2
1
2

1
  
r '2 (r '2 sin  ' dr' d ' d ' )
0 0 r 0
 4  r '4 dr' 
0

1
2
2
2
2
2
r
'
dV
'
,
where
r
'

x
'

y
'

z
'
3 
4
5

1 4
I  abc a 2  x'2 dV '  b 2  y '2 dV '  abc (a 2  b 2 ) 
3 5
I
1
M (a 2  b 2 )
5
5. Surface integrals (?) (표면적분)
dxdy  dA cos , dA  sec   dxdy
‘projection of the surface to xy plane’
 
cos  n  k
 dA   sec  dxdy
 ( x, y, z )  const.
     
normal to surface
grad  ( x, y, z )  i
j
k
x
y
z
n  ( grad ) / grad
nk 
k  grad  / z

 cos
grad
grad
grad
1
1
sec  



cos n  k  / z
(
 2  2  2
) ( ) ( )
x
y
z
 / z
For z  f ( x, y ),  ( x, y, z )  z  f ( x, y ),
so

1
z
sec  
1
f
f
 ( )2  ( )2  1
cos
x
y
Example 1. Upper surface of the sphere by the cylinder
x 2  y 2  z 2  1, x 2  y 2  y  0
 ( x, y, z )  const.
 ( x, y, z)  x 2  y 2  z 2
sec  
grad 
 / z

1
1
1
(2 x) 2  (2 y ) 2  (2 z ) 2  
2z
z
1 x2  y 2
1
y y2
y 0
x 0
2
x from 0 to y  y 2
dxdy

1 x2  y 2
y from 0 to1
 / 2 sin 
2
r from 0 to sin
 from 0 /2
 
0
x 0
rdrd
1 r
2
 /2
2

 1 r
2
d
0
0
 /2
 /2
 /2
 2  ( 1  sin   1)d  2  (1  cos )d    2
2
0
0
H. W. (due 5/28)
Chapter 5
2-43
3-17, 18, 19, 20
4-4