16 MULTIPLE INTEGRALS

CHAPTER 16
MULTIPLE INTEGRALS
• Draw a basic spherical rectangular solid S and compute that its volume is approximately
ρ 2 sin φ ρ φ θ if ρ, φ, andθ are small. Calculate the volume of the solid pictured below
to be
V = ∫ β ∫ h2 (θ) ∫ f (φ,θ)
α h 1 (θ) g(φ,θ) ρ2 sin φ dρ dφ dθ
As usual, if there is a function l (ρ, φ, θ) on S, then the triple integral is
∫ β
∫ f (φ,θ)
α g(φ,θ) l (ρ, φ, θ) ρ2 sin φ dρ dφ dθ.
∫ h2 (θ)
h 1 (θ)
• Give the students some integrands and ask them which coordinate system would be most convenient for
integrating that integrand.
1. f (x, y, z) = 1/ ( x 2 + y 2) over the solid enclosed by a piece of a circular cylinder x 2 + y 2 = a
2. f (x, y, z) = e 2x2 + 2y 2 + 2z 2 over a solid between a cone and a sphere
WORKSHOP/DISCUSSION
• Discuss conversions from cylindrical and spherical to rectangular coordinates and back. For example, the
surface sin φ = √ 1 becomes the cone z = √ x 2 + y 2 , the surface r = z cos θ becomes xz = x 2 + y 2 ,and
2
the surface z = r 2 ( 1 + 2sin 2 θ ) becomes the elliptic paraboloid z = x 2 + 3y 2 .
sin φ = 1 √
2
r = z cos θ z = r 2 ( 1 − sin 2 θ )
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