Transcript Section 14.7

Section 14.7
Triple Integrals in Cylindrical and
Spherical Coordinates
In general, to convert from
rectangular triple integrals to
cylindrical
 2 g 2 ( ) h2 ( r cos , r sin )
 f ( x, y, z)dV   
Q
1
 f (r cos , r sin  , z)rdzdrd
g1 ( ) h1 ( r cos , r sin )
To visualize
a particular
order of
integration,
view the
inetgral in
terms of
sweeping
motions.
Example
Convert the integral from rectangular to cylindrical
coordinates.
2
4 x 2
 
0
0
16 x 2  y 2

0
x  y dzdydx
2
2
Example
Convert the integral from rectangular to spherical
coordinates.
2
4 x 2
 
0
0
16 x 2  y 2

0
x  y dzdydx
2
2
Triple integrals involving spheres or
cones are often easier to evaluate
in spherical coordinates
 2 2  2
2
f
(
x
,
y
,
z
)
dV

f
(

sin

cos

,

sin

sin

,

cos

)

sin ddd


Q
1 1 1
Once again, visualize the order
of integration in terms of
sweeping motions
Figure 14.68