Triple Integrals in Cylindrical and Spherical Coordinates summarizes the process of converting between rectangular, cylindrical, and spherical coordinate systems and evaluating triple integrals using these alternative coordinate systems. It provides examples of converting between coordinate systems, finding equations for surfaces in different coordinates, and evaluating triple integrals over various regions using cylindrical and spherical coordinates.

1. 1

2. 
Triple Integrals in Cylindrical Coordinates
Useful for circle-symmetrical integration
regions and integrand functions
Switch to polar coordinates for 2 of the 3
coordinates, leave the third as is
x
r cos
y
r sin
z
z
f ( x, y , z )
f (r , , z )
dx dy dz r dr d dz
Equivalent to integrate first inz , then in polar
coordinates on the projection to thexy plane
2

3. Example 1
Convert the point r , , z
rectangular
5
,3
6
to
1, 3, 2
to
4,
coordinates.
Example 2
Convert the point x, y, z
cylindrical coordinates.
3

4. Example 3
Find an equation in cylindrical coordinates for the
surface represented
z
x2
y2 .
Example 4
Find an equation in rectangular coordinates for
the surface represented by
r 3sec .
4

5. Example 5
Using cylindrical coordinates, evaluate
a2 x2
a
0
0
a2 x2 y 2
0
x2 dz d y dx ;
a 0 .
Example 6
Find the volume of the solid that is bounded
above
x2 y 2 z 2 9
2
2
x y 4.
and below by the sphere
inside the cylinder
and
5

6. 
Triple Integrals in Spherical Coordinates
Switch to spherical coordinates: radius,
longitude, latitude
x
sin cos
y
sin sin
z
cos
x2
y2
z2
2
6

7. Switch to rectangular coordinates
x2
cos
y2
z2
z
1
x2
tan
1
y2
z2
y
x
7

8. Example 7
Convert
, ,
2,
2 5
,
3 6
to rectangular
coordinates and cylindrical coordinates.
Example 8
If the rectangular coordinates of point P are 1, 3, 2
find the spherical coordinates of P.
8

9. Example 9
Find an equation in spherical coordinates for the
surface represented by the equation
x2
y2
z2.
9

10. dV
2
sin
d d d
A typical triple integral in
spherical coordinates has
the form
f x, y, z dV
G
h2
g2
h1
g1
,
,
f
, ,
2
sin
d d d
10

11. Example 10
Use spherical coordinates to find the volume of
the
x 2 y 2 z 2 4a 2
solid enclosed by the sphere
z 0.
and
the plane 11
Example
Find the volume of the solid region Q bounded by
the cone z
x2
y2
z2
x2
y2
and the sphere
2 z.
11

12. Example 12
Use spherical coordinates to evaluate
4 y2
2
0
0
8 x2 y 2
x2 y 2
z 2 dz dx d y.
12