Triple integrals in cylindrical and spherical coordinates are useful for calculating volumes and integrals of circle-symmetric and sphere-symmetric regions. In cylindrical coordinates, the coordinates are (r, θ, z) where r is the distance from the z-axis, θ is the azimuthal angle, and z is the height. In spherical coordinates, the coordinates are (ρ, φ, θ) where ρ is the radial distance, φ is the azimuthal angle, and θ is the polar angle. Examples are provided for converting between rectangular, cylindrical, and spherical coordinates as well as setting up and evaluating triple integrals in these coordinate systems.

1. 1

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

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

4. 4
Example 3
Find an equation in cylindrical coordinates for the
surface represented
2 2
.
z x y
Example 4
Find an equation in rectangular coordinates for
the surface represented by
3sec .
r

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

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

7. 7
Switch to rectangular coordinates
2 2 2
1
2 2 2
1
cos
tan
x y z
z
x y z
y
x

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

9. 9
Example 9
Find an equation in spherical coordinates for the
surface represented by the equation
2 2 2
.
x y z

10. 10
2
sin
dV d d d
A typical triple integral in
spherical coordinates has
the form
2 2
1 1
,
2
,
, ,
, , sin
G
h g
h g
f x y z dV
f d d d

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

12. 12
Example 12
Use spherical coordinates to evaluate
2 2 2
2 2
2 4 8
2
0 0
.
y x y
x y
z dz dxdy