Chapter 6: Applications of Double Integration
Section 6.2: Volume
Example 6.2.7
If and is the region bounded by the graphs of 1, , and on , calculate the volume of the region bounded above by the surface and below by the plane .
See Example 6.1.7.
Solution
Mathematical Solution
The region whose volume is to be computed is shown in Figure 6.2.7(a). The simplest iteration of the double integral that gives this volume takes the integrand as and uses the order , with :
=
If the order of integration is taken as , then the iterated integral would be the more tedious
Figure 6.2.7(a) The volume to be computed
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Context Panel: Assign Name
Solve for
Write the equation .
Context Panel: Solve≻Numerically Solve
Context Panel: Assign to a Name≻
Access the MultiInt command via the Context Panel
Write , the name of the integrand. Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Integrate≻Iterated Fill in both panes (see Figures 5.3.(1, 2)) and select "integral" for the Output
Context Panel: Evaluate Integral
Iterate in the order via the template in the Calculus palette
Calculus palette: Iterated double-integral template
Context Panel: Evaluate and Display Inline
The task template in Table 6.2.7(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 region whose volume is computed by the integral.
Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cartesian 2-D
Evaluate and Graph
Area Element
Select dAdy dxdx dy
,
Value of Integral
Bounding Curves
"Volume"
Table 6.2.7(a) Iteration in the order via visualization task-template
Each graph in Table 6.2.7(a) has had constrained scaling imposed.
Maple Solution - Coded
Install the Student MultivariateCalculus package.
Define the integrand .
Calculate the value of
Use the fsolve command for a numeric solution.
Top-level, using the Int and int commands
Use the MultiInt command from the Student MultivariateCalculus package
Use the MultiInt command with a pre-defined domain option
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