Chapter 6: Applications of Double Integration
Section 6.2: Volume
If F=5−3 x2−2 y2 and R is the interior of the triangle whose vertices are 0,0,1,0,0,1, calculate the volume of the region bounded above by the surface z=Fx,y and below by the plane z=0.
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 R takes the integrand as F and uses the order dy dx:
∫01∫01−xF dy dx = 2512
If the order of integration is taken as dx dy, then the iterated integral would be
∫01∫01−yF dx dy = 2512
use plots in
Figure 6.2.5(a) The volume to be computed
Maple Solution - Interactive
The equation of the hypotenuse of the right triangle defining the region R is found in Table 6.2.5(a).
Tools≻Load Package: Student Precalculus
Write a sequence of two lists, each list representing an endpoint of the hypotenuse.
Context Panel: Student Precalculus≻Lines And Segments≻Line≻Equation
Context Panel: Solve≻Isolate Expression for≻x
0,1,1,0→equation of liney=−x+1→isolate for xx=−y+1
Table 6.2.5(a) Obtaining the equation of the hypotenuse for the triangle defining region R
Iterated integrals by means of which the requisite volume can be found are given in Table 6.2.5(b).
Iterate in the order dy dx via the template in the Calculus palette
Calculus palette: Iterated double-integral template
Context Panel: Evaluate and Display Inline
∫01∫01−xF ⅆy ⅆx = 2512
Iterate in the order dx dy via the template in the Calculus palette
∫01∫01−yF ⅆx ⅆy = 2512
Table 6.2.5(b) Iterated double-integrals for finding the volume "under" F and over R
Maple Solution - Coded
Install the Student MultivariateCalculus package.
Define the integrand F.
F≔5−3 x2−2 y2:
Obtain the equation of the hypotenuse in region R
Apply the Line and GetRepresentation commands.
Top-level, using the Int and int commands
Use the MultiInt command from the Student MultivariateCalculus package
MultiIntF,y=0..1−x,x=0..1 = 12⁢F
Use the MultiInt command with a pre-defined domain option
MultiIntF,x,y=Region0..1,0.. 1−x = 12⁢F
MultiIntF,y,x=Region0..1,0..1−y = 12⁢F
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