Chapter 6: Applications of Double Integration
Section 6.2: Volume
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Example 6.2.5
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If and is the interior of the triangle whose vertices are , calculate the volume of the region bounded above by the surface and below by the plane .
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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 6.2.5(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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If the order of integration is taken as , then the iterated integral would be
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use plots in
module()
local F,p1;
F:=-3*x^2-2*y^2+5;
p1:=shadebetween(0,F,x=0..1,y=0..1-x,changefill=[transparency=0],orientation=[-30,80,0],axes=frame,tickmarks=[2,2,5],labels=[x,y,z]);
print(p1);
end module:
end use:
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Figure 6.2.5(a) The volume to be computed
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Maple Solution - Interactive
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The equation of the hypotenuse of the right triangle defining the region is found in Table 6.2.5(a).
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Tools≻Load Package: Student Precalculus
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Loading Student:-Precalculus
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Write a sequence of two lists, each list representing an endpoint of the hypotenuse.
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Context Panel: Student Precalculus≻Lines And Segments≻Line≻Equation
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Context Panel: Solve≻Isolate Expression for≻
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Table 6.2.5(a) Obtaining the equation of the hypotenuse for the triangle defining region
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Iterated integrals by means of which the requisite volume can be found are given in Table 6.2.5(b).
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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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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Table 6.2.5(b) Iterated double-integrals for finding the volume "under" and over
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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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Obtain the equation of the hypotenuse in region
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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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