Chapter 6: Applications of Double Integration
Section 6.2: Volume
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Example 6.2.8
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If and is the interior of the ellipse , calculate the volume of the region bounded above by the surface and below by the plane . See Example 6.1.8.
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Solution
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Mathematical Solution
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The volume to be computed is shown in Figure 6.2.8(a). Iterating in the order , results in the integral
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Iterating in the order results in an outer integral that Maple can only evaluate numerically. This is not an uncommon feature of double integrals. Many such examples abound in textbooks whereby iteration in one order but not the other can be evaluated in closed form.
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>
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module()
local F,Y,p;
F:=sqrt(-x^2-4*y^2+2);
Y:=sqrt(-x^2+1)/2;
p:=plot3d(F,x=-1..1,y=-Y..Y,filled=true,scaling=constrained,axes=frame, labels=[x,y,z],lightmodel=none,glossiness=0,tickmarks=[3,2,3],orientation=[-60,65,0],style=surfacecontour);
print(p);
end module:
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Figure 6.2.8(a) The volume to be computed
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Maple Solution - Interactive
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The simplest approach is to employ the task template in Table 6.2.8(a).
Tools≻Tasks≻Browse:
Calculus - Vector≻Integration≻Multiple Integration≻2-D≻Over an Ellipse
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Integrate over an Ellipse
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Equation of ellipse:
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From to
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Table 6.2.8(a) Task template for integration over an ellipse
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Maple elects to implement the integration in polar coordinates, representing the ellipse as
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which is what would be obtained by the following "direct" conversion to polar coordinates.
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Expression palette: Evaluation template
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Context Panel: Solve≻Obtain Solutions for≻
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A solution from first principles is given in Table 6.2.8(b).
Initialize
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Tools≻Load Package: Student Multivariate Calculus
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Loading Student:-MultivariateCalculus
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Context Panel: Assign Name
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Solve for
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Control-drag (or type) the equation of the ellipse.
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Context Panel: Solve≻Obtain Solutions for≻
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Context Panel: Assign to a Name≻
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Access the MultiInt command via the Context Panel
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Write , the name of the integrand.
Context Panel: Evaluate and Display Inline
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Context Panel: Student Multivariate Calculus≻Integrate≻Iterated
Fill in both panes (see Figures 5.3.(1, 2)) and select "integral" for the Output
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Context Panel: Evaluate Integral
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Write an appropriate iterated integral and evaluate
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Calculus palette: Iterated double-integral template
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Context Panel: Simplify≻Simplify
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Table 6.2.8(b) Solution from first principles
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Table 6.1.8(c) provides an alternate solution via a visualization task template.
Tools≻Task≻Browse:
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cartesian 2-D
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Evaluate and Graph
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Area Element
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Value of Integral
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Table 6.2.8(c) Solution by visualization task template
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Constrained scaling has to be applied to both graphs in Table 6.2.8(c).
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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 as .
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Use the MultiInt command with a pre-defined domain option
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Alternate solution(s)
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Apply the solve command to the equation of the ellipse, obtaining the separate branches .
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Top-level, using the Int and int commands
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Iterate in the opposite order using top-level int and the evalf commands
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