16 MULTIPLE INTEGRALS

16.7 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES
TRANSPARENCY AVAILABLE
#52 (Figure 7)
SUGGESTED TIME AND EMPHASIS
1
2
–1 class Essential material
POINTS TO STRESS
1. The cylindrical coordinate system as an extension of polar coordinates in R 2 .
2. The basic shapes of cylindrical solids.
3. The idea that the cylindrical coordinate system can be used to simplify equations and volume integrals of
certain three-dimensional surfaces and solids.
QUIZ QUESTIONS
• Text Question: Does the region of Example 3 have an axis of symmetry? If so, what does it say about
the choice of using cylindrical coordinates?
Answer: The solid is symmetric about the z-axis, which implies that cylindrical coordinates should be
considered.
• Drill Question: Describe in your own words the surface given by the equation r = θ, 0≤ θ ≤ 6π in
cylindrical coordinates.
Answer: It looks something like a rolled-up piece of paper.
MATERIALS FOR LECTURE
• Compute the intersection of the surfaces z = x 2 + y 2 and z = x, first in rectangular coordinates, then in
cylindrical coordinates. (Here is a case where the rectangular coordinates are the easiest to visualize, even
though there is an x 2 + y 2 term.)
• Point out that while
lim
(x,y)→(0,0)
this, use cylindrical coordinates to get
approaches 0, as seen in Section 14.2.
xy
x 2 does not exist,
+ y2 lim
(x,y,z)→(0,0,0)
xyz
x 2 + y 2 does exist. To see
+ z2 xyz
x 2 + y 2 + z 2 = r2 z sin θ cos θ
r 2 + z 2 , and compare to
r 2 z
r 2 + z 2 ,which
• Convert a typical cylindrical volume integral of a solid S computed using double integrals into a triple
integral:
V = ∫∫ R f (r, θ) rdrdθ = ∫ β∫ h2 (θ)
α h 1 (θ) f (r, θ) rdrdθ = ∫ β∫ h2 (θ) ∫ f (r,θ)
α h 1 (θ) 0
rdzdrdθ
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