Chapter 6: Applications of Double Integration
Section 6.1: Area
Use the double integral to calculate the area of the region R, the interior of the triangle whose vertices are 0,0,a,0,0,b.
The region R is shaded in the graph shown in Figure 6.1.5(a). The simplest iteration of the double integral that gives the area of R takes the integrand as 1 and uses the order dy dx:
∫0a∫0b 1−x/a1 dy dx = a b2
If the order of integration is taken as dx dy, then the iterated integral would be
∫0b∫0a⁢1−y/b1 dx dy = a b2
Figure 6.1.5(a) The region R
Maple Solution - Interactive
The equation of the hypotenuse of the right triangle defining the region R is found in Table 6.1.5(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,b,a,0→equation of liney=−b⁢xa+b→isolate for xx=a⁢b−yb
Table 6.1.5(a) Obtaining the equation of the hypotenuse for the triangle defining region R
Tools≻Load Package: Student Multivariate Calculus
Access the MultiInt command via the Context Panel
Type the integrand, 1.
Context Panel: Student Multivariate Calculus≻Integrate≻Iterated
Fill in the fields of the two dialogs shown below
Context Panel: Evaluate Integral
From first principles, the iterated integrals by means of which the area of R can be found are given in Table 6.1.5(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
∫0a∫0b 1−x/a1 ⅆy ⅆx = 12⁢b⁢a
Iterate in the order dx dy via the template in the Calculus palette
∫0b∫0a 1−y/b1 ⅆx ⅆy = 12⁢b⁢a
Table 6.1.5(b) Iterated double-integrals for finding the area of region R
Maple Solution - Coded
Install the Student MultivariateCalculus package.
Obtain the equation of the hypotenuse in region R
Apply the isolate command to the equation returned by the Line and GetRepresentation commands.
Top-level, using the Int and int commands
Int1,y=0..b 1−x/a,x=0..a=int1,y=0..b 1−x/a,x=0..a
Int1,x=0..a 1−y/b,y=0..b=int1,x=0..a 1−y/b,y=0..b
Use the MultiInt command from the Student MultivariateCalculus package
MultiInt1,y=0..b 1−x/a,x=0..a = 12⁢b⁢a
Use the MultiInt command with a pre-defined domain option
MultiInt1,x,y=Region0..a,0.. b 1−x/a = 12⁢b⁢a
MultiInt1,x,y=Region0..a,0.. b 1−x/a,output=integral
MultiInt1,y,x=Region0..b,0..a 1−y/b = 12⁢b⁢a
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