Math 251 Sections 13.9/13.10 Triple Integrals in Cylindrical and Spherical Coordinates
If two of the variables x , y , z are converted to cylindrical coordinates and the third left as is, this is called cylindrical coordinates .
If x , y are converted to polar, then

E f ( x , y , z ) dV
 
E f ( r cos

, r sin

, z ) rdzdrd

Examples: Use cylindrical coordinates to evaluate:
1. 
E x
2  y
2 dV where E is the region bounded by z

9
 x
2  y
2 and z

0 .
2.

E xz dV where E is the region in the first octant bounded by x
2  y
2 
1 and z
 y .
Spherical Coordinates
If we convert P ( x , y , z ) to the cylindrical form P ( r ,

, z ) we have dV
 r dzdrd

.
If then the orthogonal coordinates r and z are converted to a new polar form in ρ and φ, with φ the
 the angle between OP and the positive z -axis and r
2  z
2 z
  cos

, r
  sin
 and
  2 then dV
 r dzdrd
   sin

(
 d
 d

) d
   2 sin
 d
 d
 d

In summary: x
 r cos
   sin
 cos
 y
 r sin
   sin
 sin
 z
  cos
 x
2  y
2  z
2   dV
  2 sin
 d
 d
 d

Note that the range of φ is [0, π] and the range of θ is [0, 2π]. r ,
 
0

Examples: Convert each point or equation to spherical coordinates.
1 .
2 .
a ) P ( 2 , 2 , 0 ) a ) x
2  z
2 y
2

3 b ) Q ( 2 , 2 , 2 6 ) b ) x
2  z y
2
 c ) R ( 2 , 2 ,

2 6 )
3
3 .
a ) z
 x
2  y
2 b ) z

4 x
2 
4 y
2
4 .
a ) x
2  y
2  z
2 
9 b ) x
2  y
2  z
2 
6 z
5 .
Compare cylindrical and spherical conversions for the paraboloid z
 x
2  y
2
.
6 .
Find the x
2  y
2  z
2
.
volume inside the sphere x
2  y
2  z
2  a
2 and above the cone
7 .
# 10 in 13 .
10 in Stewart Find the volume of the solid x
2  y
2  ax cuts out of the sphere x
2  y
2  z
2  a
2
.
that the cylinder
8 .
Find the centroid of the region z
 x
2  y
2 and z

36
 x
2  y
2
.
bounded by the two paraboloid s