Chapter 7: Triple Integration
Section 7.6: Integration in Spherical Coordinates
Section 5.6 details changing coordinates in a double integral. Extending this discussion to the triple integral leads to the "formula"
∫∫∫Rfx,y,z dv = ∫∫∫R′fxa,b,c,ya,b,c,za,b,c ∂x,y,z∂a,b,c dv′
where R is a region in Cartesian xyz-space, R′ is its image under the invertible mapping defined by the equations
∂x,y,z∂a,b,c = |xaxbxcyaybyczazbzc|
is the Jacobian of the transformation from R′ to R.
In particular, for spherical coordinates defined by the equations
x=ρ cosθsinφ,y=ρ sinθsinφ,z=ρ cosφ
the relevant Jacobian is ρ2sinφ. (See Example 7.6.9 for the explicit calculation of this Jacobian.)
Table 7.6.1, analogous to Table 7.3.1 for Cartesian coordinates, lists the six possible iterations for a triple integral in spherical coordinates.
∫θ=aθ=A∫φ=φθφ=Φθ∫ρ=ρφ,θρ=Ρφ,θfρ,φ,θ ρ2sinφ dρ dφ dθ
∫φ=bφ=B∫θ=θφθ=Θφ∫ρ=ρφ,θρ=Ρφ,θfρ,φ,θ ρ2sinφ dρ dθ dφ
∫θ=aθ=A∫ρ=ρθρ=Ρθ∫φ=φρ,θφ=Φρ,θfρ,φ,θ ρ2sinφ dφ dρ dθ
∫ρ=cρ=C∫θ=θρθ=Θρ∫φ=φρ,θφ=Φρ,θfρ,φ,θ ρ2sinφ dφ dθ dρ
∫φ=bφ=B∫ρ=ρφρ=Ρφ∫θ=θρ,φθ=Θρ,φfr,φ,θ ρ2sinφ dθ dρ dφ
∫ρ=cρ=C∫φ=φρφ=Φρ∫θ=θρ,φθ=Θρ,φfρ,φ,θ ρ2sinφ dθ dφ dρ
Table 7.6.1 In spherical coordinates, the six iterations of a triple integral
As in Table 7.3.1, lower-case letters are used for lower limits of integration; upper-case for upper limits. The names used for the functions in the limits of integration pertain to just the cell in which a particular iteration is displayed. Thus, in one cell the function Φρ,θ might appear, while in another Φρ might appear. The function name is pertinent only to the cell in which it appears.
Table 7.6.2 lists the basic Maple tools for iterating a triple integral in Cartesian coordinates.
∫x1x2∫y1y2∫z1z2fⅆzⅆyⅆx, the iterated triple-integral template in the Calculus palette
The Jacobian ρ2sinφ must be included in the integrand.
In the Student MultivariateCalculus package, the MultiInt command with the option "coordinates = spherical" if the coordinate names are r,φ,θ or "coordinates = spherical[n1,n2,n3]" if the coordinate names are n1,n2,n3. (Note the defaulting to r rather than to ρ.)
The task template at Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Spherical
For iteration in the order dρ dφ dθ, with exactly those variable names:
The task template at Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Multiple Integration≻Spherical
The Int and int commands at top-level
The Jacobian ρ2sinφ must be included in the integral
Table 7.6.2 Maple tools for iterating a triple integral in spherical coordinates
In Examples 7.6.(1 - 8), use spherical coordinates to integrate the function f=1 over the given region R.
R is the region above the plane z=3/4 in a unit sphere with center at the origin.
(This region was graphed in Example 7.5.3.)
R is the region above the cone z=x2+y2 but below the unit sphere that is centered at 0,0,1.
(This region was graphed in Example 7.5.4.)
(See Example 7.4.11 where this integral is evaluated in cylindrical coordinates.)
R is the region inside the unit sphere centered at the origin, and outside the cone z=x2+y2.
(This region was graphed in Example 7.5.5.)
R is the region bounded below by the cone φ=π/6 and above by the sphere ρ=2.
(This region was graphed in Example 7.5.6.)
R is the region between the spheres ρ=1 and ρ=2, and above the cone φ=π/3.
(This region was graphed in Example 7.5.7.)
R is the region above the cone φ=π/4 and below the sphere ρ=3 cosφ.
(This region was graphed in Example 7.5.8.)
R is the region enclosed by the surface ρ=1−cosφ.
(This region was graphed in Example 7.5.9.)
R is the region bounded inside by the surface ρ=1+cosφ and outside by the sphere ρ=2.
(This region was graphed in Example 7.5.10.)
If x=ρ cosθsinφ,y=ρ sinθsinφ,z=ρ cosφ, show that ∂x,y,z∂ρ,φ,θ=ρ2sinφ.
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