高等数学微积分课件7.4.2 三重积分的计算(直角坐标)

Iterative integral

{( x, y, z ) | ( x, y ) D, z1 ( x, y ) z z2 ( x, y )}z z 2 ( x, y )

If :

z z1 ( x, y )

D

z z 2 ( x, y )

z z1 ( x, y )D

( x, y)

z z 2 ( x, y )

Mass of the rod

z z1 ( x, y )D

( x, y)

f ( x, y, z )dV [D

z2 ( x , y )

z1 ( x , y )

f ( x, y, z )dz ]dxdy

f ( x, y, z )dV dxdy z ( x, y )D1

z2 ( x , y )

f ( x, y, z )dz

f ( x, y, z )dV

dxdy D

z2 ( x , y )

z1 ( x , y )

f ( x, y, z )dz

x, y视为常数“先一后二”

如果 D 是 X 型区域：

D {( x, y ) | a x b, y1 ( x) y y2 ( x)}则三重积分可以进一步化为三次积分：

f ( x, y, z )dV

dxdy D

z2 ( x , y )

z1 ( x , y )

f ( x, y, z )dzz2 ( x , y )

dx a

b

y2 ( x )

y1 ( x )

dy

z1 ( x , y )

f ( x, y, z )dz

f ( x, y, z )dV

dx a

b

y2 ( x ) y1 ( x )

dy

z2 ( x , y )

z1 ( x , y )

f ( x, y, z )dz

积分次序：

z z 2 ( x, y )

z, y , xa

z z1 ( x, y )

b

D y y1 ( x)

y y2 ( x )

Similarly, if

D {( x, y ) | c y d , x1 ( y ) x x2 ( y )}Then

f ( x, y, z )dV

dxdy D

z2 ( x , y )

z1 ( x , y )

f ( x, y, z )dzz2 ( x , y )

dy c

d

x2 ( y )

x1 ( y )

dx

z1 ( x , y )

f ( x, y, z )dz

f ( x, y, z )dV

dy c

d

x2 ( y )

x1 ( y )

dx

z2 ( x , y )

z1 ( x , y )

f ( x, y, z )dz

First with respect to z ,then x, then y

z z 2 ( x, y )

cD

z z1 ( x, y )dx x1 ( y ) x x2 ( y )

Example Solution

zdxdydz

Sketch the region of integration

1

z 1 x y

zdxdydz

dx0

1

1 x

0

dy

1 x y

0

zdz

1

D

1

y 1 x

0

1

zdxdydz 0 dx 01

1 x

dy

1 x y

0

zdz

1 x 1 1 2 1 x y dx [ z ]0 dy 0 2 0 1 x 1 1 2 dx (1 x y ) dy 0 2 0 1 1 3 1 x [(1 x y ) ]0 dx 6 0 1 1 1 1 4 1 3 (1 x) dx [(1 x) ]0 24 24 6 0

Example

xy

2 3

z dxdydz

x, y 0

z xy

bounded by

z xy y x x 1 z 0

solution: The project of onto the z axisBoundary surface z 0y xD

Boundary surface z xy

D1

z xy

D

xy z dxdydz 2 3

dx 0

1

x

0

dy

xy

0

xy z dz

2 3

1 2 4 xy dx [ xy z ]0 dy z xy 0 0 4 y x x 1 1 D dx x 5 y 6 dy D 0 4 0 1 1 1 1 5 7 x 1 1 12 1 [ x y ]0 dx x dx 4 0 7 28 0 3641

x

Another wayIf

{( x, y, z) | c z d ,( x, y) D( z)}then

f ( x, y, z )dV

[ f ( x, y, z )d ]dzc

d

D( z )

dzc

d

D( z)

f ( x, y , z ) d

f ( x,

y, z )dV

d

c

dz f ( x , y , z ) d D( z)

dz

D( z )

c

Mass of the lamina