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Chapter 7: Triple Integration
Section 7.3: Regions with Curved Boundaries
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Example 7.3.9
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If is that portion of the first-octant lying under the cylinder , but in front of the plane , iterate the triple integral in the order .
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Solution
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Mathematical Solution
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The region is shown in Figures 7.3.9(a, b), one a view from the left; and the other, from the right. Figure 7.3.9(c) is a schematic of the region . The intersection of the plane and the cylinder is drawn in green in this figure. The projection of the green curve onto the -plane is drawn in black, while its projection onto the -plane is drawn in red.
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The green curve of intersection, the simultaneous solution of the equations and , is given parametrically by equations of the form . The projection of the green curve onto the -plane is found by eliminating from the equations and . Since , the projection easily follows. The black curve is the boundary of the cylinder, but it is also the projection of the green curve of intersection onto the -plane.
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Figure 7.3.9(a) Region
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Figure 7.3.9(b) Region
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Figure 7.3.9(c) Region
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The region has a physical meaning. The cylinder can be interpreted as a piece of quarter-round molding commonly used as wood-working trim and edging. When such trim is installed in a right-angled corner, at least one end must be mitered, or cut at a bevel angle. (The saw blade would be in the plane .) A piece of quarter-round with a square end, so beveled, wastes a chip in the shape of region . The triple integral in this example computes the volume of such a cut-off scrap.
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The relevant iteration of the triple integral is . Figures 7.3.9(d - f) contain animations illustrating how the limits of integration might be deduced.
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use plots,plottools in
module()
local a,s1,s2,s3,p1,p2,p3,p4,p5,p6,p7,k;
s1 := [seq([k/30,0,sqrt(1-(k/30)^2)],k=0..30)]:
s2 := [seq([k/30,k/30,sqrt(1-(k/30)^2)],k=0..30)]:
s3 := [seq([0,k/30,sqrt(1-(k/30)^2)],k=0..30)]:
p1 := spacecurve({s1,[1,y,0,y=0..1]}, color=black, thickness=3):
p2:=spacecurve([x,x,0,x=0..1.2],color=green,thickness=2):
p3:=spacecurve(s2,color=green,thickness=2):
p4 := spacecurve({[x,1,0,x=0..1],s3}, color=red, thickness=2):
p5 := spacecurve({[[1.2,0,0], [0,0,0], [0,1.2,0]], [[0,0,0], [0,0,1.2]]}, color=blue, thickness=2):
p6:=display([p1,p2,p3,p4,p5], labels=[x,y,z], scaling = constrained, axes=none);
a:=.1;
p7:=animate(display,[parallelepiped([a,0,0],[0,a,0],[0,0,a],[.5,b,.4])],b=0..0.4,view=[0..1.2,0..1.2,0..1.2],labels=[x,y,z],orientation=[10,50,0],paraminfo=false,scaling=constrained,background=p6,tickmarks=[3,3,5]);
print(p7);
end module:
end use:
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Figure 7.3.9(d)
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use plots,plottools in
module()
local a,s1,s2,s3,p1,p2,p3,p4,p5,p6,p7,k;
s1 := [seq([k/30,0,sqrt(1-(k/30)^2)],k=0..30)]:
s2 := [seq([k/30,k/30,sqrt(1-(k/30)^2)],k=0..30)]:
s3 := [seq([0,k/30,sqrt(1-(k/30)^2)],k=0..30)]:
p1 := spacecurve({s1,[1,y,0,y=0..1]}, color=black, thickness=3):
p2:=spacecurve([x,x,0,x=0..1.2],color=green,thickness=2):
p3:=spacecurve(s2,color=green,thickness=2):
p4 := spacecurve({[x,1,0,x=0..1],s3}, color=red, thickness=2):
p5 := spacecurve({[[1.2,0,0], [0,0,0], [0,1.2,0]], [[0,0,0], [0,0,1.2]]}, color=blue, thickness=2):
p6:=display([p1,p2,p3,p4,p5], labels=[x,y,z], scaling = constrained, axes=none);
a:=.1;
p7:=animate(display,[parallelepiped([a,0,0],[0,b,0],[0,0,a],[b,0,.4])],b=0..1,view=[0..1.2,0..1.2,0..1.2],labels=[x,y,z],orientation=[10,50,0],paraminfo=false,scaling=constrained,background=p6,tickmarks=[3,3,5]);
print(p7);
end module:
end use:
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Figure 7.3.9(e)
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>
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use plots,plottools in
module()
local a,s1,s2,s3,p1,p2,p3,p4,p5,p6,p7,k;
s1 := [seq([k/30,0,sqrt(1-(k/30)^2)],k=0..30)]:
s2 := [seq([k/30,k/30,sqrt(1-(k/30)^2)],k=0..30)]:
s3 := [seq([0,k/30,sqrt(1-(k/30)^2)],k=0..30)]:
p1 := spacecurve({s1,[1,y,0,y=0..1]}, color=black, thickness=3):
p2:=spacecurve([x,x,0,x=0..1.2],color=green,thickness=2):
p3:=spacecurve(s2,color=green,thickness=2):
p4 := spacecurve({[x,1,0,x=0..1],s3}, color=red, thickness=2):
p5 := spacecurve({[[1.2,0,0], [0,0,0], [0,1.2,0]], [[0,0,0], [0,0,1.2]]}, color=blue, thickness=2):
p6:=display([p1,p2,p3,p4,p5], labels=[x,y,z], scaling = constrained, axes=none);
a:=.1;
p7:=animate(plot3d,[z,x=0..sqrt(1-z^2),y=0..x],z=0..1,view=[0..1.2,0..1.2,0..1.2],labels=[x,y,z],orientation=[10,50,0],paraminfo=false,scaling=constrained,background=p6,tickmarks=[3,3,5]);
print(p7);
end module:
end use:
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Figure 7.3.9(f)
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In Figure 7.3.9(d), the volume element first moved in the -direction, from to the "saw blade" at . This forms the "post" animated in Figure 7.3.9(e). The "post" next moves in the -direction, from to , the bounding curve in black drawn in the -plane. This forms the "slab" animated in Figure 7.3.9(f). The "slab" moves in the -direction, from to .
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Once the inner integration takes place, and the equivalent of the "post" has been formed, the base of the "post" must traverse the projection of onto the plane of the remaining two variables. Here, those two variables are and , so the region that the end of the post must traverse is the quarter-circle in the -plane.
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Maple Solution - Interactive
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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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Table 7.3.9(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.
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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 7.3.9(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 7.3.9(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 7.3.9(b) Solution via iterated triple-integral template in the Calculus palette
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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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