Chapter 6: Applications of Double Integration
Section 6.1: Area
Use the double integral to calculate the area of the region R, the region bounded by the graphs of 1, cosx, and y=x on 0≤x≤1.
The shading of the region R in Figure 6.1.7(a) supports integration in the order dy dx. However, the simplest iteration of the double integral that gives the area of R takes the integrand as 1 and uses the order dx dy, with Q≐0.74:
∫Q1∫arccosyy1 dx dy = 0.09951138789
If the order of integration is taken as dy dx, then the iterated integral would be the more tedious
∫0Q∫cosx11 dy dx+∫Q1∫x11 dy dx = 0.09951138789
Figure 6.1.7(a) The region R
Maple Solution - Interactive
Context Panel: Assign Name
Solve cosx=x for Q
Write the equation cosx=x.
Context Panel: Solve≻Numerically Solve
Context Panel: Assign to a Name≻Q
cosx=x→solve0.7390851332→assign to a nameQ
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
∫Q1∫arccosyy1 ⅆx ⅆy = 0.09951138789
Iterate in the order dy dx via the template in the Calculus palette
∫0Q∫cosx11 ⅆy ⅆx+∫Q1∫x11 ⅆy ⅆx = 0.09951138789
The task template in Table 6.1.7(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.
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 6.1.7(a) Iteration in the order dx dy via visualization task-template
Each graph in Table 6.1.7(a) has had constrained scaling imposed.
Maple Solution - Coded
Install the Student MultivariateCalculus package.
Calculate the value of Q
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
MultiInt1,x=arccosy..y,y=Q..1 = 0.09951138789
Use the MultiInt command with a pre-defined domain option
MultiInt1,y,x=RegionQ..1,arccosy..y = 0.09951138789
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