Chapter 6: Applications of Double Integration
Section 6.2: Volume
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Example 6.2.3
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If and is the region bounded by the graphs of and on , calculate the volume of the region bounded above by the surface and below by the plane . See Example 6.1.3.
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Solution
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Mathematical Solution
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The region whose volume is to be computed is shown in Figure 6.2.3(a). The simplest iteration of the double integral that gives the area of takes the integrand as F and uses the order :
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It takes two iterated double integrals to obtain the requisite area because the curves "cross" at .
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>
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use plots in
module()
local F,p1,p2,p3;
F:=2*x+3*y+1;
p1:=plot3d(F,x=0..Pi/3,y=sin(x)..sin(2*x),filled=true,lightmodel=none);
p2:=plot3d(F,x=Pi/3..Pi,y=sin(2*x)..sin(x),filled=true,lightmodel=none);
p3:=display(p1,p2,orientation=[-140,60,0],labels=[x,y,z],tickmarks=[4,3, 4]);
print(p3);
end module:
end use:
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Figure 6.2.3(a) The volume to be computed
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Maple Solution - Interactive
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Initialize
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Context Panel: Assign Name
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Obtain the intersections of the curves bounding
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Write a sequence of the two equations defining the bounding curves.
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Context Panel: Solve≻Solve
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Iterate in the order via the template in the Calculus palette
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Calculus palette: Iterated double-integral template
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Context Panel: Evaluate and Display Inline
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Context Panel: Approximate≻10 (digits)
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=
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Maple Solution - Coded
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Initialize
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Install the Student MultivariateCalculus package.
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Define the integrand .
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Top-level, using the Int and int commands
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Use the MultiInt command from the Student MultivariateCalculus package
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Use the MultiInt command with a pre-defined domain option
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