Chapter 5 Multiple integrals
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Transcript Chapter 5 Multiple integrals
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.’
22 x
(a)
2 2 x
2 2 x
y2
y0zdy y0(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
y0 x0
3
x 0
1
(b)
2
1 y / 2
1 y / 2
zdxdy
(
1
y
)
dx
dy
x
(
1
y
)
dy
A
y 0 x0
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
xa
y y1 ( x )
b
Integrate with respect to x first,
A
x2 ( y )
f ( x, y )dxdy f ( x, y )dxdy
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)dxdy
xa
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 z0dz dydx x0 y0(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 22 x
x(1 y) (1 y) / 2dydx
2
x 0 y 0
x
2
3
{(
3
2
x
)
1
}
1
/
6
{(
3
2
x
)
1
}
x0 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 xds : 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 2yds
1
A
2yds
x 0
1
2
2
2
x
1
4
x
dx
x 0
(d) centroid of surface
1
x A xdA x 2yds
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 r0 (/ r2 cos )(rdrd ) r0 r/ 2 cos drd r02r 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
zz
dV rdrddz
ds2 dr2 r 2 d 2 dz2
- Spherical coordinate
x r sin cos
y r sin sin
z r cos
dV r 2 sin drdd
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 drdd
Example 2.
z ? and I z ?
z
r=h
Mass:
2
z2
h3
M dV rdrddz 2 dz
2
3
z 0 r 0 0
0
Centroid:
h
z
h
z dV zdV
h
z
y
2
h
zrdrddz
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 rdrddz 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 drdd
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 drdd
0 0 r 0
a5 4
8a 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 drdd
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
nk
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