1. Multiple Integrals 12

2. 12.8
Triple Integrals in Cylindrical and Spherical Coordinates

3. Cylindrical Coordinates

4. Cylindrical Coordinates
Recall that the cylindrical coordinates of a point P are (r, , z), where r, , and z are shown in Figure 1.
Figure 1

5. Cylindrical Coordinates
Suppose that E is a type 1 region whose projection D onto the xy-plane is conveniently described in polar coordinates (see Figure 2).
Figure 2

6. Cylindrical Coordinates
In particular, suppose that f is continuous and
E = {(x, y, z) | (x, y)  D, u1(x, y)  z  u2(x, y)}
where D is given in polar coordinates by
D = {(r,  ) |     , h1( )  r  h2( )}
We know that

7. Cylindrical Coordinates
But we also know how to evaluate double integrals in polar coordinates. In fact, combining Equation 1 with the equation below,
we obtain

8. Cylindrical Coordinates
Formula 2 is the formula for triple integration in cylindrical coordinates.
It says that we convert a triple
integral from rectangular to
cylindrical coordinates by writing
x = r cos , y = r sin , leaving z
as it is, using the appropriate limits
of integration for z, r, and ,
and replacing dV by r dz dr d.
(Figure 3 shows how to remember this.)
Figure 3
Volume element in cylindrical
coordinates: dV = r dz dr d

9. Cylindrical Coordinates
It is worthwhile to use this formula when E is a solid region easily described in cylindrical coordinates, and especially when the function f (x, y, z) involves the expression x2 + y2.

10. Example 1 – Finding Mass with Cylindrical Coordinates
A solid E lies within the cylinder x2 + y2 = 1, below the plane z = 4, and above the paraboloid z = 1 – x2 – y2. (See Figure 4.) The density at any point is proportional to its distance from the axis of the cylinder. Find the mass of E.
Figure 4

11. Example 1 – Solution
In cylindrical coordinates the cylinder is r = 1 and the paraboloid is z = 1 – r2, so we can write
E = {(r, , z) | 0    2, 0  r  1, 1 – r2  z  4}
Since the density at (x, y, z) is proportional to the distance from the z-axis, the density function is
f (x, y, z) = K
= Kr
where K is the proportionality constant.

12. Example 1 – Solution
cont’d
Therefore, from Formula , the mass of E is

13. Example 1 – Solution
cont’d

14. Spherical Coordinates

15. Spherical Coordinates
We have defined the spherical coordinates (, , ) of a point (see Figure 6) and we demonstrated the following relationships between rectangular coordinates and spherical coordinates:
x =  sin  cos  y =  sin  sin  z =  cos 
Figure 6
Spherical coordinates of P

16. Spherical Coordinates
In this coordinate system the counterpart of a rectangular box is a spherical wedge
E = {(, , ) | a    b,     , c    d }
where a  0 and  –   2. Although we defined triple integrals by dividing solids into small boxes, it can be shown that dividing a solid into small spherical wedges always gives the same result.
So we divide E into smaller spherical wedges Eijk by means of equally spaced spheres  = i, half-planes  = j, and half-cones  = k.

17. Spherical Coordinates
Figure 7 shows that Eijk is approximately a rectangular box with dimensions , i  (arc of a circle with radius i, angle ), and i sin k  (arc of a circle with radius i sin k, angle  ).
Figure 7

18. Spherical Coordinates
So an approximation to the volume of Eijk is given by
()  (i )  (i sin k ) = i2 sin k   
Thus an approximation to a typical triple Riemann sum is
But this sum is a Riemann sum for the function
F(, , ) = f ( sin  cos ,  sin  sin ,  cos )  2 sin 

19. Spherical Coordinates
Consequently, the following formula for triple integration in spherical coordinates is plausible.

20. Spherical Coordinates
Formula 4 says that we convert a triple integral from rectangular coordinates to spherical coordinates by writing
x =  sin  cos  y =  sin  sin  z =  cos 
using the appropriate limits of
integration, and replacing dV
by  2 sin  d d d.
This is illustrated in Figure 8.
Figure 8
Volume element in spherical
coordinates: dV =  2 sin  d d d

21. Spherical Coordinates
This formula can be extended to include more general spherical regions such as
E = {(, , ) |     , c    d, g1(, )    g2(, )}
In this case the formula is the same as in (4) except that the limits of integration for  are g1(, ) and g2(, ).
Usually, spherical coordinates are used in triple integrals when surfaces such as cones and spheres form the boundary of the region of integration.

22. Example 3 Evaluate where B is the unit ball:
B = {(x, y, z) | x2 + y2 + z2  1}
Solution:
Since the boundary of B is a sphere, we use spherical coordinates:
B = {(, , ) | 0    1, 0    2, 0    }
In addition, spherical coordinates are appropriate because
x2 + y2 + z2 =  2

23. Example 3 – Solution
cont’d
Thus (4) gives