13.9 - 13.10 Part I: Triple Integrals in Cylindrical Coordinates
Cylindrical Coordinates
In the cylindrical coordinate system, a point P in R3 is represented by the ordered
triple (r, θ, z), where r and θ are polar coordinates of the projection of P onto the
xy -plane and z is the directed distance from the xy -plane to P .
To convert from cylindrical to rectangular coordinates, we use the equations
x = r cos θ,
y = r sin θ,
z=z
whereas to convert from rectangular to cylindrical coordinates, we use
r 2 = x2 + y 2 ,
tan θ =
y
,
x
z=z
Example 1. Plot the point with cylindrical coordinates (2, 2π/3, 8) and nd its rectangular coordinates.
Example
√ √ 2. Find the cylindrical coordinates of the point with rectangular coordinates
(− 2, 2, −3).
Example 3. Find an equation in cylindrical coordinates for the cone z =
p
x2 + y 2 .
Evaluating Triple Integrals with Cylindrical Coordinates
Suppose E is a region that lies between the graphs of two functions z = φ1 (x, y) and
z = φ2 (x, y), that is,
E = { (x, y, z) | (x, y) ∈ D, φ1 (x, y) ≤ z ≤ φ2 (x, y) }
where D is the projection of E onto the xy -plane and is given in polar coordinates by
D = { (r, θ) | α ≤ θ ≤ β, h1 (θ) ≤ r ≤ h2 (θ)}
Then
˚
¨ "ˆ
f (x, y, z) dV =
E
#
φ2 (x,y)
f (x, y, z) dz dA
φ1 (x,y)
D
ˆ
β
ˆ
h2 (θ) ˆ φ2 (r cos θ,r sin θ)
=
f (r cos θ, r sin θ, z)rdzdrdθ
α
Remark.
the z -axis.
h1 (θ)
φ1 (r cos θ,r sin θ)
Cylindrical coordinates are useful in problems that involve symmetry about
Example 4. The density at any point of the solid
E = { (x, y, z) | x2 + y 2 ≤ 9, −1 ≤ z ≤ 4 }
equals to its distance from the axis of E . Find the mass of E .
Example 5. Find the volume of the solid E bounded by the surfaces
y = x,
and satisfying y ≥ 0.
y = −x,
x2 + y 2 = 5z,
z=7