16 MULTIPLE INTEGRALS

SECTION 16.7
TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES
Move the bottom surface of S up to
z = g (r, θ), as pictured at right.
The volume now becomes
∫ β
∫ h2 (θ) ∫ f (r,θ)
α h 1 (θ) g(r,θ)
rdzdrdθ. The basic
volume element is given in Figure 3 of
the text. Conclude with the situation
where we have h (r, θ, z) defined on S.
Then the triple integral of h on S is
∫ β ∫ h2 (θ) ∫ f (r,θ)
α h 1 (θ) g(r,θ)
h (r, θ, z) rdzdrdθ.
WORKSHOP/DISCUSSION
• Describe in terms of cylindrical coordinates the surface of rotation formed by rotating z = 1/x about the
z-axis, noting that there is an axis of symmetry. Point out that the equations come from simply replacing
x by r.
• Develop a straightforward example such as the region depicted below:
The Capped Cone
Set this volume up as a triple integral in cylindrical coordinates, and then find the volume. (The
computation of this volume integral is not that hard, and can be assigned to the students.) Conclude
by setting up the volume integral of h (r, θ, z) = rz over this region.
GROUP WORK: A Partially Eaten Sphere
Notice that you are removing “ice cream cones” both above and below the xy-plane.
Answers:
1. 36π − 2 ∫ 2π
0
2. 36π − 2 ∫ 2π
0
∫ √ 3 ∫ √ (
6
0 −r √ 2 rdzdrdθ = π 36 − 10 √ )
6
∫ √ 3 ∫ √ 6
0 −r √ 2 zr2 sin θ dzdr dθ = 36π
913