Chapter 6: Applications of Double Integration
Section 6.1: Area
Use the double integral to calculate the area of the region R, the finite region bounded by the graph of y=x 1−x and the x-axis.
The region R is shaded in the graph shown in Figure 6.1.1(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:
∫01∫0x 1−x1 dy dx = 16
If the order of integration is taken as dx dy, then the iterated integral would be the more difficult
∫01/4∫1 −1−4 y/21+1−4 y/21 dx dy = 16
Figure 6.1.1(a) The region R
Maple Solution - Interactive
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
A solution from first principles:
Context Panel: Assign Name
Solve y=fx for x=xy
Write the equation y=fx and press the Enter key.
Context Panel: Solve≻Obtain Solutions for≻x
Context Panel: Assign to a Name≻X
(Thus, XL=X2 and XR=X1.)
→solutions for x
→assign to a name
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∫0f1 ⅆy ⅆx = 16
Iterate in the order dx dy via the template in the Calculus palette
∫01/4∫X2X11 ⅆx ⅆy = 16
The task template in Table 6.1.1(a) iterates in the order dy dx. The upward-pointing arrow in the left-hand graph indicates that the inner (first) integration is in the y-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.
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cartesian 2-D
Evaluate ∬RΨx,y dA and Graph R
Area Element dA
Select dAdy dxdx dy
Value of Integral
Table 6.1.1(a) Iteration in the order dy dx via visualization task-template
The task template in Table 6.1.1(b) iterates in the order dx dy. The right-pointing arrow in the left-hand graph indicates that the inner (first) integration is in the x-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.
Table 6.1.1(b) Iteration in the order dx dy via visualization task-template
Maple Solution - Coded
Install the Student MultivariateCalculus package.
Top-level, using the Int and int commands
Int1,y=0..x 1−x,x=0..1=int1,y=0..x 1−x,x=0..1
Use the MultiInt command from the Student MultivariateCalculus package
MultiInt1,y=0..x 1−x,x=0..1 = 16
Use the MultiInt command with pre-defined domain option
MultiInt1,x,y=Region0..1,0.. x 1−x = 16
MultiInt1,x,y=Region0..1,0.. x 1−x,output=integral
MultiInt1,y,x=Region0..1/4,1−1−4 y2..1+1−4 y2 = 16
MultiInt1,y,x=Region0..1/4,1−1−4 y2..1+1−4 y2,output=integral
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