Chapter 6: Applications of Double Integration
Section 6.1: Area
Example 6.1.2
Use the double integral to calculate the area of the region R, the finite region bounded by the graphs of x=y2 and y=3−2 x.
Solution
Mathematical Solution
The region R is shaded in the graph shown in Figure 6.1.2(a). The simplest iteration of the double integral that gives the area of R takes the integrand as 1 and uses the order dx dy:
∫−3/21∫y23−y/21 dx dy = 12548
If the order of integration is taken as dx dy, then the iterated integral is more difficult:
∫01∫−xx1 dy dx+∫19/4∫−x3−2 x1 dy dx = 12548
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:
Figure 6.1.2(a) The region R
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.
Maple Solution - Interactive
Initialize
Context Panel: Assign Name
XL=y2→assign
XR=3−y/2→assign
Obtain the intersections of the curves bounding R
Write a sequence of the two equations defining the bounding curves.
Context Panel: Solve≻Solve
x=y2,y=3−2 x→solvex=1,y=1,x=94,y=−32
Iterate in the order dx dy via the template in the Calculus palette
Calculus palette: Iterated double-integral template
Context Panel: Evaluate and Display Inline
∫−3/21∫y23−y/21 ⅆx ⅆy = 12548
Iterate in the order dy dx via the template in the Calculus palette
∫01∫−xx1 ⅆy ⅆx+∫19/4∫−x3−2 x1 ⅆy ⅆx = 12548
The task template in Table 6.1.2(a) 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.
Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cartesian 2-D
Evaluate ∬RΨx,y dA and Graph R
Area Element dA
Select dAdy dxdx dy
, Ψ=
Value of Integral
G=
b=
g=
a=
Bounding Curves
"Volume"
Table 6.1.2(a) Iteration in the order dx dy via visualization task-template
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Top-level, using the Int and int commands
Int1,x=y2..3−y/2,y=−3/2..1=int1,x=y2..3−y/2,y=−3/2..1
∫−321∫y232−12y1ⅆxⅆy=12548
Use the MultiInt command from the Student MultivariateCalculus package
MultiInt1,x=y2..3−y/2,y=−3/2..1 = 12548
MultiInt1,x=y2..3−y/2,y=−3/2..1,output=integral
∫−321∫y232−12y1ⅆxⅆy
MultiInt1,x=y2..3−y/2,y=−3/2..1,output=steps
12548
Use the MultiInt command with a pre-defined domain option
MultiInt1,y,x=Region−3/2..1,y2..3−y/2 = 12548
MultiInt1,y,x=Region−3/2..1,y2..3−y/2,output=integral
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