Chapter 7: Triple Integration
Section 7.4: Integration in Cylindrical 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 cylindrical coordinates defined by the equations
x=r cosθ,y=r sinθ,z=z
the relevant Jacobian is r. (See Example 7.4.15 for the explicit calculation of this Jacobian.)
Table 7.4.1, analogous to Table 7.3.1 for Cartesian coordinates, lists the six possible iterations for a triple integral in cylindrical coordinates.
∫θ=aθ=A∫r=rθr=Rθ∫z=zr,θz=Zr,θfr,θ,z r dz dr dθ
∫r=br=B∫θ=θrθ=Θr∫z=zr,θz=Zr,θfr,θ,z r dz dθ dr
∫z=cz=C∫θ=θzθ=Θz∫r=rθ,zr=Rθ,zfr,θ,z r dr dθ dz
∫θ=aθ=A∫z=zθz=Zθ∫r=rθ,zr=Rθ,zfr,θ,z r dr dz dθ
∫z=cz=C∫r=rzr=Rz∫θ=θr,zθ=Θr,zfr,θ,z r dθ dr dz
∫r=br=B∫z=zrz=Zr∫θ=θr,zθ=Θr,zfr,θ,z r dθ dz dr
Table 7.4.1 In cylindrical 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 Zr,θ might appear, while in another Zr might appear. The function name is pertinent only to the cell in which it appears.
Table 7.4.2 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 r must be included in the integrand.
In the Student MultivariateCalculus package, the MultiInt command with the option "coordinates = cylindrical" if the coordinate names are r,θ,z or "coordinates = cylindrical[n1,n2,n3]" if the coordinate names are n1,n2,n3.
The task template at Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cylindrical
For iteration in the order dz dr dθ:
The task template at Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Multiple Integration≻Cylindrical
The Int and int commands at top-level
The Jacobian r must be included in the integral
Table 7.4.2 Maple tools for iterating a triple integral in cylindrical coordinates
Table 7.4.3 illustrates the syntax by means of which the plot3d command will graph a surface defined in cylindrical coordinates.
plot3dfn1,n2,n1=range 1, n2=range 2, coords=cylindrical
plot3dru,v,θu,v,zu,v,u=range u, v=range v,coords=cylindrical
Table 7.4.3 Syntax by which the plot3d command will graph in cylindrical coordinates
In the explicit case, the first range and its coordinate name is interpreted as the angle θ; the second, as z. The function f is interpreted as rθ,z.
In the parametric case, the parameters u and v could be any two of r,θ,z. For example, if u=r and v=θ, then the parametric list would be r,θ,zr,θ.
In Examples 7.4.(1 - 14), use cylindrical coordinates to integrate the function f=1 over the given region R.
R is the region bounded above by the upper hemisphere in a sphere of radius 1, and below by the plane through the "equator".
R is the interior of the cylinder x2+y2=1/4, bounded above by the plane x+y+z=1, and below by the plane z=0.
R is the region bounded below by z=x2+y2, above by z=1+x2+y2, and laterally by the cylinder x2+y2=1.
R is the region bounded above by z=5−x2−y2 and below by z=4 x2+4 y2.
R is the region bounded above by the sphere x2+y2+z2=2, and below by z=x2+y2.
R is the region outside the cylinder x2+y2=1, bounded above by z=9−x2−y2, and below by z=0.
R is the region above z=1+3 x2+3 y2, below 2 x+3 y+ z=11, and inside the cylinder x2+y2=1.
R is the region bounded below by the paraboloid z=3 x2+3 y2 and above by the plane z=5.
R is the region inside the cylinder whose cross section is the cardioid r=3+2 cosθ, bounded above by the plane x+2 y+3 z=10, and below by the plane z=0.
R is that part of the interior of the sphere x2+y2+z2=4 that lies inside the cylinder whose cross section is r=2 sinθ.
R is the region above the cone z=x2+y2 but below the unit sphere that is centered at 0,0,1.
R is the region above the plane z=0, below the cone z=2x2+y2, and inside the cylinder whose cross section is the cardioid r=3+2 sinθ.
R is the region above the plane z=0, below the plane z=x, inside the cylinder whose cross section is x2+y2=1, and lying in the half-space x≥0.
R is the region between the planes z=0 and 2 x+3 y+5 z=17, and above the annulus whose radii are 1 and 2, and whose center is at the origin.
If x=r cosθ,y=r sinθ, z=z, show that ∂x,y,z∂r,θ,z=r.
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