16 MULTIPLE INTEGRALS

16.8 TRIPLE INTEGRALS IN SPHERICAL COORDINATES
TRANSPARENCY AVAILABLE
#52 (Figure 7)
SUGGESTED TIME AND EMPHASIS
1–2 classes Essential material
POINTS TO STRESS
1. The geometry of the spherical coordinate system.
2. The basic shapes of solids in spherical coordinates.
3. The idea that the spherical coordinate system can be used to simplify equations and volume integrals of
certain three-dimensional surfaces and solids.
QUIZ QUESTIONS
• Text Question: What surface is given by the equation ρ = 3?
Answer: A sphere
• Drill Question: What is the solid described by the integral ∫ π ∫ 2π ∫ √ 3
π/2 0 0
ρ 2 sin ϕ dρ dθ dϕ?
Answer: The bottom half of a sphere with radius √ 3
MATERIALS FOR LECTURE
• One way to demonstrate spherical coordinates is to sit in a pivoting office chair, holding a yardstick or,
better yet, an extendable pointer. Now ρ can be demonstrated in the obvious way, θ by spinning in the
chair, and ϕ by raising and lowering the yardstick. The instructor can do this, or students can come up and
be asked to “touch” certain points with the ruler.
• Describe the coordinates of all points 3 units from the origin in each of the three coordinate systems.
Repeat for the coordinates of all points on a circular cylinder of radius 2 with central axis the z-axis, and
the coordinates of all points on the line through the origin with direction vector i + j + k. Conclude
that cylindrical coordinates are the most useful in problems that involve symmetry about an axis, and that
spherical coordinates are most useful where there is symmetry about a point.
• Identify the somewhat mysterious surface ρ cos φ = ρ 2 sin 2 φ cos 2θ given in spherical coordinates by
using the formula cos 2θ = cos 2 θ − sin 2 θ and changing to rectangular coordinates.
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