Chapter 7: Triple Integration
Section 7.3: Regions with Curved Boundaries
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Example 7.3.2
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If is that portion of the first-octant lying under the cylinder for , iterate the triple integral in the order .
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Solution
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Mathematical Solution
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The required iteration is
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The animations in Figures 7.3.2(b - d) indicate how the element of volume (a small parallelepiped) must move so as to sweep out the region .
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According to the animation in Figure 7.3.2(b), the element of volume must move in the -direction first, from to the surface defined by .
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Figure 7.3.2(a) Region
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The result of this inner integration is to produce the "post" animated in Figure 7.3.2(c). This "post" must move in the -direction, from to , resulting in the "slab" that is animated in Figure 7.3.2(d). This "slab" now must move in the -direction, from to .
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use plots, plottools in
module()
local a,p1,p2;
a:=.2;
p1:=plot3d(4-x^2,x=0..2,y=0..3);
p2:=animate(display,[parallelepiped([a,0,0],[0,a,0],[0,0,a],[1,1.5,b])],b=0..3,view=[0..2,0..3,0..4],labels=[x,y,z],orientation=[-130,75,0],paraminfo=false,scaling=constrained,background=p1,tickmarks=[3,3,5]);
print(p2);
end module:
end use:
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Figure 7.3.2(b)
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use plots, plottools in
module()
local a,p1,p2;
a:=.2;
p1:=plot3d(4-x^2,x=0..2,y=0..3);
p2:=animate(display,[parallelepiped([a,0,0],[0,a,0],[0,0,4-b^2],[b,1.5,0])],b=0..2,view=[0..2,0..3,0..4],labels=[x,y,z],orientation=[-130,75,0],paraminfo=false,scaling=constrained,background=p1,tickmarks=[3,3,5]);
print(p2);
end module:
end use:
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Figure 7.3.2(c)
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use plots, plottools in
module()
local a,p1,p2;
a:=.2;
p1:=plot3d(4-x^2,x=0..2,y=0..3);
p2:=animate(plot3d,[[x,b,z],x=0..2,z=0..4-x^2],b=0..3,view=[0..2,0..3,0..4],labels=[x,y,z],orientation=[-130,75,0],paraminfo=false,scaling=constrained,background=p1,tickmarks=[3,3,5]);
print(p2);
end module:
end use:
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Figure 7.3.2(d)
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Maple Solution - Interactive
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Table 7.3.2(a) contains a solution provided by a visualization task template. After the order of iteration is selected, fill in the fields that correspond to the limits of integration. If the graph of the region swept by these limits is correct, then the integral is correctly formulated and evaluated.
Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cartesian 3-D
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Evaluate and Graph
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Volume Element
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, where
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Table 7.3.2(a) Solution by visualization task template
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This template employs the MultiInt command from the Student MultivariateCalculus package, but the graphic are coded from first principles.
Table 7.3.2(b) contains a solution implemented with the iterated triple-integral template found in the Calculus palette.
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Calculus palette: Iterated triple-integral template
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Context Panel: Evaluate and Display Inline
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Table 7.3.2(b) Solution via iterated triple-integral template in the Calculus palette
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Access the MultiInt command via the Context Panel
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Context Panel: Student Multivariate Calculus≻Integrate≻Iterated
Fill in the fields of the two dialogs shown below
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Context Panel: Evaluate Integral
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Maple Solution - Coded
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Top-level: Int and int commands
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The MultiInt command in the Student MultivariateCalculus package
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Install the Student MultivariateCalculus package.
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The MultiInt command with a pre-defined domain option
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