James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 15.7 Triple Integrals in Cylindrical Coordinates 1041
ExamplE 1
(a) Plot the point with cylindrical coordinates s2, 2y3, 1d and find its rectangular
coordinates.
(b) Find cylindrical coordinates of the point with rectangular coordinates s3, 23, 27d.
z
SOLUTION
(a) The point with cylindrical coordinates s2, 2y3, 1d is plotted in Figure 3. From
Equations 1, its rectangular coordinates are
2π
”2, 3 , 1’
1
x − 2 cos 2 3 − 2 S2 1 2D − 21
0
x
FIGURE 3
2π
3
2
y
y − 2 sin 2 3
z − 1
− 2 S s3
2
D − s3
So the point is s21, s3 , 1d in rectangular coordinates.
(b) From Equations 2 we have
r − s3 2 1 s23d 2 − 3s2
z
tan − 23
3 − 21 so − 7 4 1 2n
(c, 0, 0)
x
FIGURE 4
r=c, a cylinder
0
(0, c, 0)
y
z − 27
Therefore one set of cylindrical coordinates is s3s2 , 7y4, 27d. Another is
s3s2 , 2y4, 27d. As with polar coordinates, there are infinitely many choices.
Cylindrical coordinates are useful in problems that involve symmetry about an axis,
and the z-axis is chosen to coincide with this axis of symmetry. For instance, the axis of
the circular cylinder with Cartesian equation x 2 1 y 2 − c 2 is the z-axis. In cylindrical
coordinates this cylinder has the very simple equation r − c. (See Figure 4.) This is the
reason for the name “cylindrical” coordinates.
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x
0
FIGURE 5
z=r , a cone
z
y
ExamplE 2 Describe the surface whose equation in cylindrical coordinates is z − r.
SOLUTION The equation says that the z-value, or height, of each point on the surface is
the same as r, the distance from the point to the z-axis. Because doesn’t appear, it can
vary. So any horizontal trace in the plane z − k sk . 0d is a circle of radius k. These
traces suggest that the surface is a cone. This prediction can be confirmed by converting
the equation into rectangular coordinates. From the first equation in (2) we have
z 2 − r 2 − x 2 1 y 2
We recognize the equation z 2 − x 2 1 y 2 (by comparison with Table 1 in Section 12.6)
as being a circular cone whose axis is the z-axis (see Figure 5).
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