Chapter 6: Applications of Double Integration
Section 6.1: Area
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Example 6.1.2
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Use the double integral to calculate the area of the region , the finite region bounded by the graphs of and .
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Solution
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Mathematical Solution
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The region is shaded in the graph shown in Figure 6.1.2(a). The simplest iteration of the double integral that gives the area of takes the integrand as 1 and uses the order :
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If the order of integration is taken as , then the iterated integral is more difficult:
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use plots in
module()
local p1,p2,p3;
p1:=shadebetween(-sqrt(x),sqrt(x),x=0..1,color=red,changefill=[color=blue,transparency=.7],thickness=3):
p2:=shadebetween(-sqrt(x),3-2*x,x=1..9/4,color=black,changefill=[color=green,transparency=.7],thickness=3):
p3:=display(p1,p2,labels=[x,y],tickmarks=[3,[-1.5,-1,0,1]]);
print(p3);
end module:
end use:
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Figure 6.1.2(a) The region
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The first of these two iterated integrals gives the area of the blue-shaded region in Figure 6.1.2(a); the second, the green.
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Maple Solution - Interactive
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Initialize
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Context Panel: Assign Name
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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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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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The task template in Table 6.1.2(a) iterates in the order . The right-pointing arrow in the left-hand graph indicates that the inner (first) integration is in the -direction. The right-hand image contains a graph of the volume computed by the integral. Since the height of the region in the graph is the constant 1, the volume and the area are the same number.
Tools≻Tasks≻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 6.1.2(a) Iteration in the order via visualization task-template
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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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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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